Evaluate the following limits :
`Lim_( x to 0) (e^(ax)-e^(bx))/x `

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x}, \]

we can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit to facilitate simplification. We can add and subtract 1 in the numerator:

\[ \lim_{x \to 0} \frac{e^{ax} - 1 - (e^{bx} - 1)}{x}. \]

### Step 2: Separate the terms
Now, we can separate the limit into two parts:

\[ \lim_{x \to 0} \left( \frac{e^{ax} - 1}{x} - \frac{e^{bx} - 1}{x} \right). \]

### Step 3: Use the derivative definition
Notice that as \( x \to 0 \), both terms resemble the derivative of \( e^{kx} \) at \( x = 0 \). We can rewrite each term using the definition of the derivative:

\[ \lim_{x \to 0} \frac{e^{ax} - 1}{x} = a \quad \text{and} \quad \lim_{x \to 0} \frac{e^{bx} - 1}{x} = b. \]

### Step 4: Combine the results
Substituting these limits back into our expression gives:

\[ a - b. \]

### Step 5: Final result
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{e^{ax} - e^{bx}}{x} = a - b. \]