Question

evaluate the limit, if it exists. (if an answer does not exist, enter dn
lim_{t
ightarrow0}\frac{sqrt{5 + t}-sqrt{5 - t}}{t}

Explanation:

Step1: Rationalize the numerator

Multiply numerator and denominator by $\sqrt{5 + t}+\sqrt{5 - t}$.
\[

$$\begin{align*} &\lim_{t ightarrow0}\frac{\sqrt{5 + t}-\sqrt{5 - t}}{t}\times\frac{\sqrt{5 + t}+\sqrt{5 - t}}{\sqrt{5 + t}+\sqrt{5 - t}}\\ =&\lim_{t ightarrow0}\frac{(5 + t)-(5 - t)}{t(\sqrt{5 + t}+\sqrt{5 - t})} \end{align*}$$

\]

Step2: Simplify the numerator

Simplify $(5 + t)-(5 - t)$ to $2t$.
\[

$$\begin{align*} &\lim_{t ightarrow0}\frac{2t}{t(\sqrt{5 + t}+\sqrt{5 - t})} \end{align*}$$

\]

Step3: Cancel out the common factor

Cancel out the common factor $t$ (since $t
eq0$ as we are taking the limit as $t$ approaches 0).
\[

$$\begin{align*} &\lim_{t ightarrow0}\frac{2}{\sqrt{5 + t}+\sqrt{5 - t}} \end{align*}$$

\]

Step4: Evaluate the limit

Substitute $t = 0$ into the expression.
\[

$$\begin{align*} &\frac{2}{\sqrt{5+0}+\sqrt{5 - 0}}=\frac{2}{2\sqrt{5}}=\frac{1}{\sqrt{5}} \end{align*}$$

\]

Answer:

$\frac{1}{\sqrt{5}}$