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Try this beautiful problem on Calculus, useful for ISI B.Stat Entrance.

## Problem on Calculus | ISI B.Stat Entrance | Problem 696

If k is an integer such that lim $\{{cos}^n(k\pi/4) – {cos}^n(k\pi/6)\} = 0$,
then

• (a) k is divisible neither by 4 nor by 6
• (b) k must be divisible by 12, but not necessarily by 24
• (c) k must be divisible by 24
• (d) either k is divisible by 24 or k is divisible neither by 4 not by 6

### Key Concepts

Calculus

Limit

Trigonometry

But try the problem first…

Source

TOMATO, Problem 694

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

There are four options ,at first we have to check each options…..

If k is divisible by 24 then cos(kπ/4) = cos(kπ/6) = 1
$\Rightarrow$ The limit exists and equal to RHS i.e. 0
If k is not divisible by 4 or 6 then cos(kπ/4), cos(kπ/6) both <1

Can you now finish the problem ……….

Final Step

Therefore ,

lim cosn(kπ/4), cosn(kπ/6) = 0. so we may say that
$\Rightarrow$The equation holds.