Question

Evaluate: $\underset{x\to 0}{\lim }\frac{e^{5x}-e^{4x}}{x}$

• A. $1$
• B. $2$
• C. $4$
• D. $5$

Answer 1

The correct option is A

$1$

Explanation for the correct answer:

Simplifying the equation to determinate form and applying the limits:

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\frac{e^{5x}-e^{4x}}{x} \\ \Rightarrow \underset{x\to 0}{\lim }\frac{e^{5x}-1-e^{4x}+1}{x}[\text{Adding}-1+1\text{in the numerator}] \\ \Rightarrow \underset{x\to 0}{\lim }\frac{\left(e^{5x}-1\right)}{x}-\frac{\left(e^{4x}+1\right)}{x} \\ \Rightarrow \underset{x\to 0}{\lim }\frac{5\left(e^{5x}-1\right)}{5x}-\frac{4\left(e^{4x}+1\right)}{4x}[\text{Multiply and divide}5i\text{n first term and}4\text{in second term}] \end{gathered}$$

Applying the limits

$$\begin{gathered} \Rightarrow 5\times 1-4\times 1\left[\because \underset{x\to 0}{\lim }\left(\frac{e^x-1}{x}\right)=1\right] \\ \Rightarrow 5-4 \\ \Rightarrow 1 \end{gathered}$$

Thus, $\underset{x\to 0}{\lim }\frac{e^{5x}-e^{4x}}{x}=1$

Therefore, the correct answer is option (A).