Question

The given limit represents the derivative of a function f at a number a Find f and .a

$\underset{x\to 5}{\lim }\frac{2^x-32}{x-5}$

Short answer

The value of$f\left(x\right)$is$2^x$and value of a is 5.

Step-by-step solution

Step 1. Write the given expression.

The expression is $\underset{x\to 5}{\lim }\frac{2^x-32}{x-5}$.

Step 2. Write the formula used.

The instantaneous rate of change of a function, and the instantaneous velocity of an object at $x=a$ all required us to compute the following limit.$f\text{'}\left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}\mathrm{........}\left(i\right)$

Step 3. Find the value of f and a.

Compare equation (i) with the given expression. So, we get

$f\left(x\right)=2^x$

And,

$\begin{array}{l}f\left(a\right)=32\\ f\left(a\right)=2^5\\ a=5\end{array}$

Therefore, the value of $f\left(x\right)$is$2^x$ and value of is 5 .