Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically.

$\underset{x\to 0}{\lim }\frac{5^x-3^x}{x}$

Short answer

The value of limit of the given function is$\frac{1}{2}$.

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 0}{\lim }\frac{5^x-3^x}{x}$

Step 2. Apply L’Hopital’s Rule.

Then,

$\underset{x\to 0}{\lim }\left(\frac{5^x\ln \left(5\right)-3^x\ln \left(3\right)}{1}\right)$

Now, plug in the value $x=0$

Then we get,$\frac{5^0\ln \left(5\right)-3^0\ln \left(3\right)}{1}$

Step 3. Simplify.

$\frac{5^0\ln \left(5\right)-3^0\ln \left(3\right)}{1}=\ln \left(5\right)-\ln \left(3\right)$

Step 4. Make the table.

Now make a table,

Therefore, on the basis of the values in the table we make the guess that,

$\underset{x\to 0}{\lim }\frac{5^x-3^x}{x}=0.50=\frac{1}{2}$

Step 5. Draw the graph.

Now, as a graphical verification use a graphing device.

From the graph it is verified that,$\underset{x\to 0}{\lim }\frac{5^x-3^x}{x}=\frac{1}{2}$