Examples On Evaluating Limits Set-4

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Example - 8

Evaluate the following limits:

(a) \(\mathop {\lim }\limits_{x \to 0} \,\sin \frac{1}{x}\) (b) \(\mathop {\lim }\limits_{x \to 0} \,\,x\sin \frac{1}{x}\) (c) \(\mathop {\lim }\limits_{x \to + \infty } \,\,\frac{{\ln x}}{x}\)
(d) \(\mathop {\lim }\limits_{x \to {0^ + }} \,x\ln x\) (e) \(\mathop {\lim }\limits_{x \to + \infty } \,\,\,\frac{{{x^n}}}{{n!}}\)  

Solution: (a) Notice that as x \(\to\) 0, \(\frac{1}{x} \to \infty \) , that is , \(\frac{1}{x}\) has no particular limit to which it converges. Hence \(\sin \frac{1}{x}\) keeps oscillating between +1 and –1 as x becomes smaller and smaller, i.e., x → 0.

Therefore, the limit for this function does not exist.

This is also clear from the graph (approximate) of \(\sin \frac{1}{x}\) sketched below:

(b) In this limit, in addition to  \(\sin \frac{1}{x}\), ‘x’ is also present. Thus, although \(\sin \frac{1}{x}\) remains oscillating and does not approach any particular limit, it nevertheless remains somewhere between +1 and –1, and when it gets multiplied by x(where x → 0), the whole product gets infinitesimally small.

That is, 

\(\mathop {\lim }\limits_{x \to 0} \,\,x\sin \frac{1}{x} = 0\)

Again, this is evident from the graph below:

(c) This limit can be evaluated purely by observation as follow:

Although \(\ln x\) and x are both tending to infinity, \(\ln x\) increases very slowly as compared to x.

For example, when \(x{\rm{ }} = {\rm{ }}{e^{10}}\), ln x is just 10. When \(x{\rm{ }} = {\rm{ }}{e^{10000}}\)(a very large number indeed !),

\(\ln x\)  is just 10000.

Therefore, \(\frac{{\ln x}}{x}\) decreases and becomes infinitesimally small as x →\(\infty \) , i.e.,

\(\mathop {\lim }\limits_{x \to \infty } \,\,\,\frac{{\ln x}}{x} = 0\)

(We can also use the LH rule to evaluate the limit above: this rule will be discussed later)

(d) Consider  \(xlnx\).

As \(x \to {0^ + }\) , \(\ln x \to - \infty \) , so that this limit is of the indeterminate form \(0 \times \infty .\)

But as in parts (b) and (c), try to see that the product becomes infinitesimally small as \(x \to 0.\) .

For example, at \(x = {e^{ - 10}}\) , \(ln{\rm{ }}x{\rm{ }} = {\rm{ }}-10\) and \(x\ln x = \frac{{ - 10}}{{{e^{10}}}}\)

At \(x = {e^{ - 1000}}\) , \(x\ln x = \frac{{ - 1000}}{{{e^{1000}}}}\) (which is very very small)

Hence, here again, \(\mathop {\lim }\limits_{x \to {0^ + }} x\ln x = 0\)

(e) If \(\left| x \right| < 1,\) then as \(n \to \infty \) , \(Num \to 0\) and \(Den \to \infty \) , so that the limit is 0.

For \(\left| x \right| = 1,\) also, the limit is obviously 0.

For \(\left| x \right| > 1\) we write \(\frac{{{x^n}}}{{n!}}\) as

\(\frac{{{x^n}}}{{n!}} = \frac{x}{1} \cdot \frac{x}{2} \cdot \frac{x}{3}...\frac{x}{n}\)

Now, since x is finite, let N be the integer just less than or equal to x ; N = [x]

Hence,\(\begin{align}\frac{{{x^n}}}{{n!}} = \frac{x}{1} \cdot \frac{x}{2}...\frac{x}{N} \cdot \frac{x}{{N + 1}} \cdot \frac{x}{{N + 2}}...\frac{x}{n}\end{align}\)

The product of the first N terms is finite; let it be equal to P.

Thus \(\mathop {\lim }\limits_{n \to \infty } \frac{{{x^n}}}{{n!}} = P\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{x}{{N + 1}} \cdot \frac{x}{{N + 2}}...\frac{x}{n}} \right\}\)

The product inside the limit consists of all terms less than 1. Also successive terms become smaller and smaller and tend to 0 as \(n \to \infty \) .

Therefore, this product tends to 0 and hence the value of the overall limit is P × 0 = 0

\(\mathop {\lim }\limits_{n \to \infty } \frac{{{x^n}}}{{n!}} = 0\)

Note: As we mentioned earlier, once we have studied differentiation, we’ll study the L’Hospital’s rule for evaluation of limits of the form \(\frac{0}{0}\;{\rm{or}}\;\frac{\infty }{\infty }\) . However, it might be useful to know the rule right away - so we provide a brief idea here:

Here are two examples:

(i) \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{{x^3}}} = \mathop {\lim }\limits_{x \to 0} \frac{{{{\sec }^2}x - 1}}{{3{x^2}}}{\rm{ }}\left( {\;{\rm{still}}\frac{0}{0}\;} \right)\)

\( = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sec }^2}x}}{6}\;\left( {\frac{{\tan x}}{x}} \right)\)

\(= \frac{2}{6} = \frac{1}{3}\)

(ii) \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( {\pi {{\cos }^2}x} \right)}}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \frac{{\cos \left( {\pi {{\cos }^2}x} \right)}}{2}\begin{array}{*{20}{c}}{\; \times - 2\pi \cos x}\\\end{array}\;\left( {\frac{{\sin x}}{x}} \right)\)

\(= - \frac{1}{2} \times - 2\pi\)

\( = \pi\)

This rule is simple yet extremely powerful, and in general, you’ll be able to solve most limits using this rule.

TRY YOURSELF – II

‘Limits’ is a subject where lots of practice is required. That is why this exercise is long! Make sure you solve all these questions before moving ahead.

Evaluate the following limits:

1. \(\begin{align}\mathop {\lim }\limits_{x \to 2} \frac{{{x^3} - 6{x^2} + 11x - 6}}{{{x^2} - 6x + 8}}\end{align}\) . 2. \(\begin{align}\mathop {\lim }\limits_{x \to 3} \frac{{{x^3} - 7{x^2} + 15x - 9}}{{{x^4} - 5{x^3} + 27x - 27}}.\end{align}\)
3. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \left( {\frac{1}{{1 - x}} - \frac{3}{{1 - {x^3}}}} \right)\end{align}\) 4. \(\begin{align}\mathop {\lim }\limits_{x \to 2} \left( {\frac{2}{{x\left( {x - 2} \right)}} - \frac{1}{{{x^2} - 3x + 2}}} \right) \end{align}\)
5. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \left( {\frac{{x + 2}}{{{x^2} - 5x + 4}} + \frac{{x - 4}}{{3\left( {{x^2} - 3x + 2} \right)}}} \right) \end{align}\) 6.  \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right) - 1}}{x} \end{align}\)
7. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{{x^4} - 3x + 2}}{{{x^5} - 4x + 3}} \end{align}\) 8. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {1 + x} \right)}^5} - \left( {1 + 5x} \right)}}{{{x^2} + {x^5}}} \end{align}\)
9. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \left\{ {\left( {1 + x} \right)\left( {1 + {x^2}} \right)\left( {1 + {x^4}} \right)...\left( {1 + {x^{{2^n}}}} \right)} \right\},\,\,{\rm{if }}\left| x \right| < 1\end{align}\) 10. \(\begin{align}\mathop {\lim }\limits_{x \to a} \frac{{\sqrt {a + 2x} - \sqrt {3x} }}{{\sqrt {3a + x} - 2\sqrt x }}\end{align}\)
11. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {{x^2} + 1} - 1}}{{\sqrt {{x^2} + 16} - 4}}\end{align}\) 12. \(\begin{align}\mathop {\lim }\limits_{x \to \sqrt {10} } \frac{{\sqrt {7 + 2x} - \left( {\sqrt 5 + \sqrt 2 } \right)}}{{{x^2} - 10}}\end{align}\)
13. \(\begin{align}\mathop {\lim }\limits_{x \to \sqrt 6 } \frac{{\sqrt {5 + 2x} - \sqrt 3 - \sqrt 2 }}{{{x^2} - 6}}\end{align}\) 14. \(\begin{align}\mathop {\lim }\limits_{x \to \sqrt 2 } \frac{{\sqrt {3 + 2x} - \sqrt 2 - 1}}{{{x^2} - 2}}\end{align}\)
15. \(\begin{align}\mathop {\lim }\limits_{x \to 4} \frac{{\sqrt {1 + 2x} - 3}}{{\sqrt x - 2}}\end{align}\) 16. \(\begin{align}\mathop {\lim }\limits_{x \to a} \frac{{{x^m} - {a^m}}}{{{x^n} - {a^n}}}.\end{align}\)
17. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{\left( {x + {x^2} + {x^3} + ... + {x^n}} \right) - n}}{{x - 1}}.\end{align}\) 18. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {1 + x} \right)}^5} - 1}}{{3x + 5{x^2}}}\end{align}\)
19. \(\begin{align}\mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {x + h} \right)}^{1/n}} - {x^{1/n}}}}{h}\end{align}\) 20. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{{x^2} - \sqrt x }}{{\sqrt x - 1}}\end{align}\)
21. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{x + {{\left( {x + 1} \right)}^2} + x{{\left( {x + 1} \right)}^3} + ... + {{\left( {x + 1} \right)}^n} - n + 1}}{x}\end{align}\) 22. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{\left( {1 - x} \right)\left( {1 - {x^2}} \right)...\left( {1 - {x^{2n}}} \right)}}{{{{\left[ {\left( {1 - x} \right)\left( {1 - {x^2}} \right)...\left( {1 - {x^n}} \right)} \right]}^2}}}\end{align}\)
23. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{a{x^2} + bx + c}}{{d{x^2} + ex + f}}.\end{align}\) 24. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {3{x^2} - 1} - \sqrt {2{x^2} - 1} }}{{4x + 3}}\end{align}\)
25. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \sqrt {{x^2} + x + 1} - \sqrt {{x^2} + 1} .\end{align}\) 26. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {n + 1} \right)}^4} - {{\left( {n - 1} \right)}^4}}}{{{{\left( {n + 1} \right)}^4} + {{\left( {n - 1} \right)}^4}}}\end{align}\)
27. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {2n + 1} \right)}^4} - {{\left( {n - 1} \right)}^4}}}{{{{\left( {2n + 1} \right)}^4} + {{\left( {n - 1} \right)}^4}}}\end{align}\) 28. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{\sqrt[3]{{{n^3} + 2n - 1}}}}{{n + 2}}\end{align}\)
29. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{\left( {n + 1} \right)! - n!}}\end{align}\) 30. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{\sqrt {{n^3} - 2{n^2} + 1} + \sqrt {{n^4} + 1} }}{{\sqrt[4]{{{n^6} + 2{n^5} + 2}} - \sqrt[5]{{{n^7} + 3{n^3} + 1}}}}\end{align}\)
31. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{\sqrt[4]{{{n^5} + 2}} - \sqrt[3]{{{n^2} + 1}}}}{{\sqrt[5]{{{n^4} + 2}} - \sqrt {{n^3} + 1} }}\end{align}\) 32. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{\left( {n + 2} \right)! + \left( {n + 1} \right)!}}{{\left( {n + 3} \right)!}}\end{align}\)
33. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{\left( {n + 2} \right)! + \left( {n + 1} \right)!}}{{\left( {n + 2} \right)! - \left( {n + 1} \right)!}}\end{align}\) 34. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{1.2}} + \frac{1}{{2.3}} + .... + \frac{1}{{\left( {n - 1} \right)n}}} \right]\end{align}\)
35. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{1.3}} + \frac{1}{{3.5}} + .... + \frac{1}{{\left( {2n - 1} \right)\left( {2n + 1} \right)}}} \right]\end{align}\) 36. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^3}}}{{2{x^2} - 1}} - \frac{{{x^2}}}{{2x + 1}}} \right)\end{align}\)
37. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{3{x^2}}}{{2x + 1}} - \frac{{\left( {2x - 1} \right)\left( {3{x^2} + x + 2} \right)}}{{4{x^2}}}} \right)\end{align}\) 38. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{{{\left( {x + 1} \right)}^{10}} + {{\left( {x + 2} \right)}^{10}} + ... + {{\left( {x + 100} \right)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}\end{align}\)
39. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {{x^2} + 1} + \sqrt x }}{{\sqrt[4]{{{x^3} + x}} - x}}\end{align}\) 40. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {{x^2} + 1} - \sqrt[3]{{{x^2} + 1}}}}{{\sqrt[4]{{{x^4} + 1}} - \sqrt[5]{{{x^4} + 1}}}}\end{align}\)
41. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt[5]{{{x^7} + 3}} + \sqrt[4]{{2{x^3} - 1}}}}{{\sqrt[6]{{{x^8} + {x^7} + 1}} - x}}\end{align}\) 42. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt[3]{{{x^4} + 3}} - \sqrt[5]{{{x^3} + 4}}}}{{\sqrt[3]{{{x^7} + 1}}}}\end{align}\)
43. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \sqrt[3]{{{{\left( {x + 1} \right)}^2}}} - \sqrt[3]{{{{\left( {x - 1} \right)}^2}}}\end{align}\) 44. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \sqrt[4]{{\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right)\left( {x + d} \right)}} - x\end{align}\)
45. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt x }}{{\sqrt {x + \sqrt {x + \sqrt x } } }}\end{align}\) 46. \(\begin{align}\mathop {\lim }\limits_{x \to \pi /6} \frac{{\sin \left( {x - \pi /6} \right)}}{{\sqrt 3 /2 - \cos x}}\end{align}\)
47. \(\begin{align}\mathop {\lim }\limits_{\alpha \to \beta } \frac{{{{\sin }^2}\alpha - {{\sin }^2}\beta }}{{{\alpha ^2} - {\beta ^2}}}\end{align}\) 48. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \left( {\cos \sqrt {x + 1} - \cos \sqrt x } \right)\end{align}\)
49. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{{2^{x - 1}} - 1}}{{\sin \pi x}}\end{align}\) 50. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{{e^{{x^2}}} - \cos x}}{{{x^2}}}\end{align}\)
51. \(\begin{align}\mathop {\lim }\limits_{x \to \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}},a > 0\end{align}\) 52. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{x - {{\log }_e}\left( {1 + x} \right)}}{{{x^2}}}\end{align}\)
53. \(\begin{align}\mathop {\lim }\limits_{x \to e} \frac{{{{\log }_e}x - 1}}{{x - e}}\end{align}\) 54. \(\begin{align}\mathop {\lim }\limits_{x \to 2} \frac{{x - 2}}{{{{\log }_a}\left( {x - 1} \right)}}\end{align}\)
55. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{{{27}^x} - {9^x} - {3^x} + 1}}{{\sqrt 2 - \sqrt {1 + \cos x} }}\end{align}\) 56. \(\begin{align}\mathop {\lim }\limits_{x \to \pi /2} \frac{{{a^{2x}} - {a^\pi }}}{{\cos x}}\end{align}\)
57. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {1 + \sin x} \right)^{2\cot x}}\end{align}\) 58. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\cos x} \right)^{\cot x}}\end{align}\)
59. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x} + {c^x}}}{3}} \right)^{1/x}}\end{align}\) 60. \(\begin{align}\mathop {\lim }\limits_{x \to a} {\left( {2 - \frac{a}{x}} \right)^{\tan \pi x/2a}}\end{align}\)

61. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left\{ {\tan \left( {\frac{\pi }{4} + x} \right)} \right\}^{1/x}}\end{align}\)

62. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{\sin x}}{x}} \right)^{1/{x^2}}}\end{align}\)
63. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{x + 5}}{{x - 1}}} \right)^x}\end{align}\) 64. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{{x^2} + 4x - 3}}{{{x^2} - 2x + 5}}} \right)^x}\end{align}\)
65. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } {\left\{ {\sin \frac{1}{x} + \cos \frac{1}{x}} \right\}^x}\end{align}\) 66. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left\{ {\frac{{{1^x} + {2^x} + .... + {n^x}}}{n}} \right\}^{a/x}}\end{align}\)
67. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\cos x + \sin x} \right)^{1/x}}\end{align}\) 68. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\cos x} \right)^{1/\sin x}}\end{align}\)
69. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\cos x + a\sin bx} \right)^{1/x}}\end{align}\) 70. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{3x - 4}}{{3x + 2}}} \right)^{\frac{{x + 1}}{3}}}\end{align}\)
71. \(\begin{align}\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{\tan x}}{x}} \right)^{1/x}}\end{align}\) 72. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } {\left\{ {\frac{{3{x^2} + 1}}{{4{x^2} - 1}}} \right\}^{\frac{{{x^3}}}{{1 + x}}}}\end{align}\)
73. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}} + ... + \frac{{n - 1}}{{{n^2}}}} \right).\end{align}\) 74. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{{n^2} + 1}} + \frac{2}{{{n^2} + 1}} + .... + \frac{{n - 1}}{{{n^2} + 1}}} \right).\end{align}\)
75. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{2{n^2} + n - 1}}{{\left( {n + 1} \right) + \left( {n + 2} \right) + ... + 2n}}.\end{align}\) 76. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{{n^2}}}{{{2^n}}}.\end{align}\)
77. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{{2^n}}}{{n!}}.\end{align}\) 78. \(\begin{align}\mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{{n^n}}}.\end{align}\)
79. \(\begin{align}\mathop {\lim }\limits_{x \to 2} \frac{{x - \sqrt {3x - 2} }}{{{x^2} - 4}}.\end{align}\) 80. \(\begin{align}\mathop {\lim }\limits_{x \to - 8} \frac{{\sqrt {1 - } x - 3}}{{2 + \sqrt[3]{x}}}.\end{align}\)
81. \(\begin{align}\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt[3]{x} - 1}}{{\sqrt x - 1}}.\end{align}\) 82. \(\begin{align}\mathop {\lim }\limits_{x \to 9} \frac{{\sqrt[3]{{x - 1}} - 2}}{{x - 9}}.\end{align}\)
83. \(\begin{align}\mathop {\lim }\limits_{x \to 16} \frac{{\sqrt[4]{x} - 2}}{{\sqrt x - 4}}.\end{align}\) 84. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + x} - \sqrt {1 - x} }}{{\sqrt[3]{{1 + x}} - \sqrt[3]{{1 - x}}}}.\end{align}\)
85. \(\begin{align}\mathop {\lim }\limits_{x \to 9} \left( {\frac{{3 - \sqrt x }}{{9 - x}} + \frac{1}{{3 - \sqrt x }} - 6.\frac{{{x^2} + 162}}{{729 - {x^3}}}} \right)\end{align}\) . 86. \(\begin{align} \mathop {\lim }\limits_{x \to 0} x\cot 3x. \end{align}\) \(\)
87. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\cos \left( {x/2} \right) - \sin \left( {x/2} \right)}}{{\cos x}}.\end{align}\) 88. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos x}}{{{x^2}}}.\end{align}\)
89. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\sin 5x - \sin 3x}}{{\sin x}}.\end{align}\) 90. \(\begin{align}\mathop {\lim }\limits_{x \to a} \frac{{\cos x - \cos a}}{{x - a}}.\end{align}\)
91. \(\begin{align}\mathop {\lim }\limits_{x \to \pi /6} \frac{{2{{\sin }^2}x + \sin x - 1}}{{2{{\sin }^2}x - 3\sin x + 1}}.\end{align}\) 92. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 - \cos {x^2}} }}{{1 - \cos x}}.\end{align}\)
93. \(\begin{align}\mathop {\lim }\limits_{x \to 0} \frac{{\sin x - \tan x}}{{{{\sin }^3}x}}.\end{align}\) 94. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2} + 3x - 4}}{{1 - 5{x^2}}}.\end{align}\)
95. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{3x}}{{5x - 1}}\,\,\frac{{2{x^2} + 1}}{{{x^2} + 2x - 1}}} \right).\end{align}\) 96. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{2{x^2} + 7x - 2}}{{6{x^3} - 4x + 3}}.\end{align}\)
97. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 4} \right)\left( {x - 5} \right)}}{{{{\left( {5x - 1} \right)}^5}}}.\end{align}\) 98. \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {{x^2} + 1} - x}}{{x + 1}}.\end{align}\)
99. \(\begin{align}\mathop {\lim }\limits_{x \to + \infty } \left( {x - \sqrt {{x^2} + 5x} } \right).\end{align}\) 100. \(\begin{align}\mathop {\lim }\limits_{x \to - 4} \frac{{\sqrt {5 + x} - 1}}{{{x^2} + 4x}}.\end{align}\)
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