Q. 16 - 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, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why.

$\underset{x\to 4}{\lim }f\left(x\right)g\left(x\right)$

Step-by-Step Solution

Answer

The value of $\underset{x\to 4}{\lim }f\left(x\right)g\left(x\right)$ cannot be calculated because $value of \underset{x\to 4}{\lim }g\left(x\right)$ is not given

Step 1. Given information

The given limit $\underset{x\to 4}{\lim }f\left(x\right)g\left(x\right)$ and given values $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Step 2. Calculating value of the given limits

The value can be calculated as below,

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

Since the value of $\underset{x\to 4}{\lim }g\left(x\right)$is not given,therefore,the value of the expression cannot be calculated