Q. 13 - 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. 

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

Step-by-Step Solution

Answer

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

Step 1. Given information

The given limits $\underset{x\to 3}{\lim }\left(2f\left(x\right)-3g\left(x\right)\right)$ and the given values are,

$\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ 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,

$$\begin{gathered} \underset{x\to 3}{\lim }\left(2f\left(x\right)-3g\left(x\right)\right)=\underset{x\to 3}{\lim }\left(2f\right(x\left)\right)-\underset{x\to 3}{\lim }\left(3g\right(x\left)\right) [sum rule] \\ =2\underset{x\to 3}{\lim }f\left(x\right)-3\underset{x\to 3}{\lim }g\left(x\right) [cons\tan t multiple rule] \\ =2\left(5\right)-3\left(4\right) \\ =10-12 \\ =-2 \end{gathered}$$

Hence,the result is $-2$