# 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}