Question

How do you find the limit of ${e^{\dfrac{1}{x}}}$ as x approaches ${0^ - }$ ?

Answer 1

Here in this question we need to find the limit for the function. Let we define the given function as $f(x)$ and we apply the limit to it. The function will get closer and closer to some number. When the x approaches to ${0^ - }$ we need to find the limit of a function.

Complete step by step explanation:
The idea of a limit is a basis of all calculus. The limit of a function is defined as let $f(x)$ be a function defined on an interval that contains $x = a$ . Then we say that $\mathop {\lim }\limits_{x \to a} f(x) = L$ , if for every $\varepsilon > 0$ there is some number
$\delta > 0$ such that $\left| {f(x) - L} \right| < \varepsilon$ whenever $0 < \left| {x - a} \right| < \delta$ .
Here we have to find the value of the limit when x is zero. Let we define the given function as $f(x) = {e^{\dfrac{1}{x}}}$ . Now we are going to apply the limit to the function $f(x)$ so we have
$\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{x \to {0^ - }} {e^{\dfrac{1}{x}}}$
The function $f(x)$ is an exponential function. As x approaches to ${0^ - }$ we have to substitute the x
as ${0^ - }$ .
When we consider the x value we will consider it as a negative value because they have given ${0^ - }$
$\Rightarrow \mathop {\lim }\limits_{x \to {0^ - }} f(x) = {e^{ - \dfrac{1}{0}}}$
Any number divided by zero then it becomes infinity, then
$\Rightarrow \mathop {\lim }\limits_{x \to {0^ - }} f(x) = {e^{ - \infty }}$
By the exponential property we have ${e^{ - \infty }} = 0$ . By considering this property we have
$\mathop {\lim }\limits_{x \to {0^ - }} f(x) = 0$
Therefore we have found the limit for the function $f(x) = {e^{\dfrac{1}{x}}}$

Hence $\mathop {\lim }\limits_{x \to {0^ - }} {e^{\dfrac{1}{x}}} = 0$

Note: The plus and minus sign refer to the direction from which the function approaches zero. If they mention ${0^ - }$ the limit as x approaches 0 from the negative side or from below. If they mention ${0^ \+ }$ the limit as x approaches 0 from the positive side or from above. The property of exponential function should be known to solve this problem.