A student graphs using two calculator windows

MATHEMATICS · Single correct · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

A student graphs

f(x)=\sin\!\left(\dfrac{1}{x}\right)

using two calculator windows. • Window A:

-5\le x\le 5

-1\le y\le 1

• Window B:

-0.1\le x\le 0.1

-1\le y\le 1

In Window A the curve appears nearly flat and seems to approach

0

x\to 0

. In Window B the curve oscillates between

-1

and

1

with increasing frequency as

x\to 0

. Which statement best describes

\displaystyle\lim_{x\to 0}f(x)

? (a) The limit does not exist because

f

is not defined at

x=0

. (b) The limit does not exist because

f

oscillates without approaching a single value near

x=0

. (c) The limit exists and equals

0

. (d) The limit does not exist because the left-hand and right-hand limits are different.