Evaluate the limits, if exist`lim_(x->3)(e^x-e^3)/(x-3)`

Explanation

To evaluate the limit

\[ \lim_{x \to 3} \frac{e^x - e^3}{x - 3}, \]

we will follow these steps:

### Step 1: Identify the Form of the Limit
First, we need to substitute \( x = 3 \) into the expression to check if it results in an indeterminate form.

\[ \frac{e^3 - e^3}{3 - 3} = \frac{0}{0}. \]

Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule or manipulate the expression.

### Step 2: Rewrite the Limit
We can rewrite the limit as follows:

\[ \lim_{x \to 3} \frac{e^x - e^3}{x - 3} = \lim_{x \to 3} \frac{e^3(e^{x-3} - 1)}{x - 3}. \]

### Step 3: Factor Out the Constant
Since \( e^3 \) is a constant, we can factor it out of the limit:

\[ = e^3 \lim_{x \to 3} \frac{e^{x-3} - 1}{x - 3}. \]

### Step 4: Change of Variable
Now, let’s make a substitution. Let \( y = x - 3 \). As \( x \to 3 \), \( y \to 0 \). Thus, we can rewrite the limit in terms of \( y \):

\[ = e^3 \lim_{y \to 0} \frac{e^y - 1}{y}. \]

### Step 5: Use the Standard Limit
We know from calculus that:

\[ \lim_{y \to 0} \frac{e^y - 1}{y} = 1. \]

### Step 6: Final Calculation
Substituting this result back into our limit gives:

\[ = e^3 \cdot 1 = e^3. \]

### Conclusion
Thus, the limit exists and is equal to:

\[ \lim_{x \to 3} \frac{e^x - e^3}{x - 3} = e^3. \]