Q. 18 - Calculus

Question

Suppose f and g are functions such that $\underset{x\to 3}{lim} f\left(x\right)=5$, $\underset{x\to 4}{lim} f\left(x\right)=2$, and $\underset{x\to 3}{lim} g\left(x\right)=4$. Given this information, calculate the limits that follow, if possible. If it is not possible with the given information, explain why. 

Step-by-Step Solution

Answer

$\underset{x\to 3}{lim} f\left(g\right(x\left)\right)=2$

Step 1. Given information.

We have to find the value of $\underset{x\to 3}{lim} f\left(g\right(x\left)\right)$

Given values are:

$$\begin{gathered} \underset{x\to 3}{lim} f\left(x\right)=5 \\ \underset{x\to 4}{lim} f\left(x\right)=2 \\ \underset{x\to 3}{lim} g\left(x\right)=4 \end{gathered}$$

Step 2. Find the value

The value can be calculated as follows:

$$\begin{gathered} \underset{x\to 3}{lim} f\left(g\right(x\left)\right) \\ =f\left(\underset{x\to 3}{lim} g\left(x\right)\right) \\ =f\left(4\right) \\ =\underset{x\to 4}{lim} f\left(x\right) \\ =2 \end{gathered}$$