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UPSC NDA I 2016 Mathematics Solved Paper

Question : 81 of 120

Marks: +1, -0

\lim\limits_{x\to 0}\phi(x)=a^2

\lim\limits_{x\to 0}\phi\left(\frac{x}{a}\right)

Equal to

Solution:

Concept:

The limit of a function as its argument approaches zero depends only on the value of the function near zero.

Explanation:

Given:

\lim\limits_{x\to 0}\phi(x)=a^2

, where

a \neq 0

We need

\lim\limits_{x\to 0}\phi\left(\frac{x}{a}\right)

Let

t = \frac{x}{a}

. Then

x = a t

x \to 0

t \to 0

because

a

is a non‑zero constant.

Thus

\lim\limits_{x\to 0}\phi\left(\frac{x}{a}\right) = \lim\limits_{t\to 0}\phi(t)

But

\lim\limits_{t\to 0}\phi(t) = \lim\limits_{x\to 0}\phi(x) = a^2

Therefore the required limit is

a^2

Answer:

a^2

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