NDA 2017 Maths · Mathematics · Limits · medium

Lim x→0x^2cos 1/x=0. 2. The function f(x)=(sin 3x)/(x) for x≠0 with f(0)=1 is continuous at x=0.

Single correct option

Consider the following statements:
1.

\displaystyle\lim_{x\to0}x^2\cos\frac1x=0

.
2. The function

f(x)=\frac{\sin3x}{x}

for

x\ne0

with

f(0)=1

is continuous at

x=0

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) $1$ only

|x^2\cos\frac1x|\le x^2\to0

by the sandwich theorem. 2:

\lim_{x\to0}\frac{\sin3x}{x}=3\ne f(0)=1

, so

f

is discontinuous at

0

. Only statement 1 is correct; the value

f(0)=3

would make

f

continuous.

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