Provide example limit equations for exponential and logarithmic functions.

Steps to Solve

1. Example 1: Limit involving exponential function

Here's a limit problem involving an exponential function. $$ \lim_{x \to 0} \frac{e^x - 1}{x} $$

2. Solve Example 1

This limit is a standard result and can be solved using L'Hôpital's rule or by recognizing it as the derivative of $e^x$ at $x=0$. Applying L'Hôpital's rule:

Since the limit is in the indeterminate form $\frac{0}{0}$, we differentiate the numerator and the denominator with respect to $x$. $$ \lim_{x \to 0} \frac{\frac{d}{dx}(e^x - 1)}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{e^x}{1} $$ Now, we can directly evaluate the limit by substituting $x = 0$: $$ \lim_{x \to 0} e^x = e^0 = 1 $$

3. Example 2: Limit involving logarithmic function

Consider the following limit involving a logarithmic function: $$ \lim_{x \to 0} \frac{\ln(1+x)}{x} $$

4. Solve Example 2

This limit is also a standard result. Similar to the previous example, we can use L'Hôpital's rule or recognize it as the derivative of $\ln(1+x)$ at $x=0$. Applying L'Hôpital's rule:

Since the limit is in the indeterminate form $\frac{0}{0}$, we differentiate the numerator and the denominator with respect to $x$. $$ \lim_{x \to 0} \frac{\frac{d}{dx}(\ln(1+x))}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{\frac{1}{1+x}}{1} $$ Now, we can directly evaluate the limit by substituting $x = 0$: $$ \lim_{x \to 0} \frac{1}{1+x} = \frac{1}{1+0} = 1 $$