Question

Consider the following:

1. \(\rm \underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\) exists.

2. \(\rm \underset{x\to 0}{\mathop{\lim }}\, e^{\frac 1 x}\) does not exist.

Which of the above is / are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
5. None of these

Answer 1

Correct Answer - Option 2 : 2 only

Concept: 

 \(\rm \lim_{x\rightarrow 0} f(x) = f(0)\)

Calculations:

Consider, \(\rm f(x) = \frac{1}{x}\) 

⇒ \(\rm \lim_{x\rightarrow 0} f(x)= \lim_{x\rightarrow 0}\frac{1}{x}\)

⇒ \(\rm \lim_{x\rightarrow 0} = \frac{1}{0} = \infty\)

\(\rm \underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\) does not exist

 

Now, \(\rm f(x) = e^{\frac{1}{x}}\)

⇒ \(\rm \lim_{x\rightarrow 0} f(x)= \lim_{x\rightarrow 0}e^{\frac{1}{x}}\)

⇒ \(\rm \lim_{x\rightarrow 0} f(x)= e^{\frac{1}{0}}\)

⇒ \(\rm \lim_{x\rightarrow 0} f(x)= e^\infty= \infty\)

Hence, \(\rm \underset{x\to 0}{\mathop{\lim }}\, e^{\frac 1 x}\) does not exist.