Evaluate the following limits in Exercises 1 to 22: limₓ→₀ (ax + x cos x)/(b sin x)

Solution:

At x = 0, the value of the given function takes the form 0/0, which is an indeterminate form.

So we will evaluate the given limit differently.

Now,

limₓ→₀ (ax + x cos x)/(b sin x)

= 1/b limₓ→₀ x(a + cos x)/(sin x)

= 1/b limₓ→₀ x/(sin x) · limₓ→₀ (a + cos x)

= 1/b · 1/[limₓ→₀ (sin x/x)] · limₓ→₀ (a + cos x)

= 1/b · 1/1 · (a + cos 0) [∵ limₓ→₀ (sin x/x) = 1]

= (a + 1)/b

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.1 Question 18

Evaluate the following limits in Exercises 1 to 22: limₓ→₀ (ax + x cos x)/(b sin x)

Summary:

The value of the limit limₓ→₀ (ax + x cos x)/(b sin x) is (a + 1)/b