Question

c) \\( \lim _ { x \
ightarrow \infty } x \sin ( 2 / x ) \\)

Explanation:

Step1: Variable substitution

Let \( t=\frac{1}{x} \). As \( x
ightarrow\infty \), \( t
ightarrow0 \).
The limit becomes \( \lim_{t
ightarrow0}\frac{\sin(2t)}{t} \).

Step2: Use the limit formula \( \lim_{u

ightarrow0}\frac{\sin u}{u} = 1 \)
We know that \( \lim_{t
ightarrow0}\frac{\sin(2t)}{t}=2\lim_{t
ightarrow0}\frac{\sin(2t)}{2t} \).
Let \( u = 2t \), when \( t
ightarrow0 \), \( u
ightarrow0 \). So \( 2\lim_{t
ightarrow0}\frac{\sin(2t)}{2t}=2\lim_{u
ightarrow0}\frac{\sin u}{u} \).
Since \( \lim_{u
ightarrow0}\frac{\sin u}{u} = 1 \), then \( 2\lim_{u
ightarrow0}\frac{\sin u}{u}=2\times1 \).

Answer:

\( 2 \)