Evaluate the following limits in Exercises 1 to 22: lim ₓ→₀ (sin ax)/(sin bx), a, b ≠ 0

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, the given limit is

lim ₓ→₀ (sin ax)/(sin bx)

Multiplying and dividing by each of ax and bx,

= lim ₓ→₀ [ [ax/bx] (sin ax/ax)/lim ₓ→₀ [(sin bx/bx)] 

When x → 0 ⇒ ax → 0 and x → 0 ⇒ bx → 0. So the above limit becomes

= (a/b) [lim ₐₓ→₀  (sin ax/ax)] / [lim bₓ→₀ (sin bx/bx) ]      

= a/b × 1/1 [∵ lim_y→0 (siny/y) = 1]

= a/b

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.1 Question 14

Evaluate the following limits in Exercises 1 to 22: lim ₓ→₀ (sin ax)/(sin bx), a, b ≠ 0

Summary:

The value of the limit lim ₓ→₀ (sin ax)/(sin bx), a, b ≠ 0 is a/b