Question

How do you find the limit of $ {10^x} $ as $ x $ approaches infinity?

Answer 1

Hint: In order to determine the limit of the above function, check whether the value is in the indeterminate form i.e. $ \dfrac{0}{0}\,or\,\dfrac{{ \pm \infty }}{{ \pm \infty }},{0^0} $ etc or not. To remove the indeterminate form, simplify the function into its simplest form then put the value of limit. If then also you get the same then check the left- hand limit and right-hand limit, if needed.

Complete step by step solution:
We are given with the function $ {10^x} $ as $ x $ approaches infinity, which in limit form can be written as:
 $ \mathop {\lim }\limits_{x \to \infty } {10^x} $
We can see that the function we have is not in fraction form, and also $ x $ approaches infinity not zero, so the value will not be in indeterminate form.
Now putting the value of $ x = \infty $ , as the function cannot be further simplified and we get:
\[\;\mathop {\lim }\limits_{x \to \infty } {10^x} = {10^{\mathop {\lim }\limits_{x \to \infty } x}} = {10^\infty }\]
Further solving the function, we get:
\[\;\mathop {\lim }\limits_{x \to \infty } {10^\infty } = \infty \], As we know that positive number with infinity, for limit approaching toward infinity, would head towards positive infinity.
Therefore, Limit for the function \[\;{10^x}\]as $ x $ approaches infinity\[\;\mathop { = \lim }\limits_{x \to \infty } {10^x} = + \infty \].
So, the correct answer is “ $ \infty $ ”.

Note: Don’t forget to cross-check your answer.
2) After putting the limit, the result should never in the indeterminate form $ \dfrac{0}{0}\,or\,\dfrac{{ \pm \infty }}{{ \pm \infty }} $ . If it is contained, apply some operation to modify the result or use the L-Hospitals rule.
3) If instead of \[x\], there was \[ - x\], then the limit would have approached towards negative infinity.