Question

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

Answer 1

Hint: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.