Question Video: Finding the Limit of a Composition of Functions | Nagwa Question Video: Finding the Limit of a Composition of Functions | Nagwa

# Question Video: Finding the Limit of a Composition of Functions Mathematics • Second Year of Secondary School

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Assume that lim_(π₯ β β1) π(π₯) = 3. If π(π₯) = 2^π₯, then find lim_(π₯ β β1) π(π(π₯)).

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### Video Transcript

Assume that the limit as π₯ approaches negative one of π of π₯ is equal to three. If π of π₯ is equal to two to the power of π₯, then find the limit as π₯ approaches negative one of π evaluated at π of π₯.

In this question, weβre told that the limit as π₯ approaches negative one of π of π₯ is equal to three and weβre also given a function π of π₯, which is two to the power of π₯. We need to use this information to determine the limit as π₯ approaches negative one of π evaluated at π of π₯. And we can start by noticing the limit weβre asked to evaluate is the limit of a composition of two functions. So, letβs start by recalling our result about evaluating the limit of the composition of two functions.

We know if we have the limit as π₯ approaches π of a function π of π₯ is equal to πΏ and we have a function π which is continuous at πΏ, then the limit as π₯ approaches π of π composed of π of π₯ is equal to π evaluated at πΏ. We can think of this as taking the limit inside of the function or we can also think of this as taking the function outside of the limit. Weβll apply this to our question. We need to know that our function π of π₯ is an exponential function, which we know is continuous for all real values of π₯.

Now to apply this to our question, we know our value of π is negative one and our value of πΏ is three. Substituting these values into our limit result, we get the limit as π₯ approaches negative one of two to the power of π of π₯ is equal to two cubed, which is just equal to eight. Therefore, we were able to show if the limit as π₯ approaches negative one of π of π₯ is equal to three and π of π₯ is equal to two to the power of π₯, then the limit as π₯ approaches negative one of π evaluated at π of π₯ is equal to eight.

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