Q. 17 - 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.

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

Step-by-Step Solution

Answer

The value of $\underset{x\to 3}{\lim }\frac{f\left(x\right)-3}{g\left(x\right)}=\frac{1}{2}$

Step 1. Given information

The given limit $\underset{x\to 3}{\lim }\frac{f\left(x\right)-3}{g\left(x\right)}$ 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,

$$\begin{gathered} \underset{x\to 3}{\lim }\frac{f\left(x\right)-3}{g\left(x\right)}=\frac{\underset{x\to 3}{\lim }f\left(x\right)-3}{\underset{x\to 3}{\lim }g\left(x\right)} [quotient rule] \\ =\frac{\underset{x\to 3}{\lim }f\left(x\right)-\underset{x\to 3}{\lim }3}{\underset{x\to 3}{\lim }g\left(x\right)} [sum rule] \\ =\frac{5-3}{4} \\ =\frac{2}{4} \\ =\frac{1}{2} \end{gathered}$$

Hence,the result is $\frac{1}{2}$