Question

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

$\underset{t\to 1}{\lim }\frac{\sqrt{t+1}-\sqrt{2}}{t-1}$

Short answer

The value of$f\left(x\right)$ is$\sqrt{t+1}$ and value of a is1.

Step-by-step solution

Step 1. Write the given expression.

The expression is $\underset{t\to 1}{\lim }\frac{\sqrt{t+1}-\sqrt{2}}{t-1}$.

Step 2. Write the formula used.

The instantaneous rate of change of a function, and the instantaneous velocity of an object at $x=a$$$" width="9" height="19" role="math">

$f\text{'}\left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}\mathrm{........}\left(i\right)$

Step 3. Find the value of f and a.

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

$f\left(x\right)=\sqrt{t+1}$

And,

$\begin{array}{l}f\left(a\right)=\sqrt{2}\\ f\left(a\right)=\sqrt{1+1}\\ a=1\end{array}$$\begin{array}{l}f\left(a\right)=\sqrt{2}\\ f\left(a\right)=\sqrt{1+1}\\ a=1\end{array}$

Therefore, the value of$f\left(x\right)$is$\sqrt{t+1}$and value of is 1 .