Evaluate the following limits in Exercises 1 to 22: lim z→₁ (z¹/³ - 1)/(z¹/⁶ - 1)

Solution:

At z = 1, the value of the given function takes the form 0/0, which is an indeterminate form.

So we will evaluate the given limit differently.

Put z^1/6 = x so that z → 1 as x → 1

Accordingly,

lim z→₁ (z^1/3 - 1)/(z^1/6 - 1) = lim ₓ→₁ (x² - 1)/(x - 1)

= lim ₓ→₁ (x² - 1²)/(x - 1)

= 2 × 1^{2 - 1} [∵ (xⁿ - aⁿ)/(x - a) = na^{n - 1}]

= 2

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.1 Question 10

Evaluate the following limits in Exercises 1 to 22: lim z→₁ (z¹/³ - 1)/(z¹/⁶ - 1)

Summary:

The value of the limit lim z→₁ (z^1/3 - 1)/(z^1/6 - 1) is 2