Find the limit (if it exists). (If an answer does not exist, enter DNE.)
3 E15
lim x+Ax x
Ax—0* AZ

Question

Find the limit (if it exists). (If an answer does not exist, enter DNE.)
3 E15
lim x+Ax x
Ax—0* AZ

VIPSolutions · Accepted Answer

Given:

$$
\lim_{\Delta x \to 0^+} \frac{\frac{3}{x + \Delta x} - \frac{3}{x}}{\Delta x}
$$

Let's solve it step by step:

### Step 1: Combine the fractions in the numerator:

$$
\frac{3}{x+\Delta x} - \frac{3}{x} = 3\left(\frac{1}{x+\Delta x} - \frac{1}{x}\right)
$$
$$
= 3\left(\frac{x - (x + \Delta x)}{x(x + \Delta x)}\right)
$$
$$
= 3\left(\frac{-\Delta x}{x(x + \Delta x)}\right)
$$
$$
= \frac{-3\Delta x}{x(x + \Delta x)}
$$

### Step 2: Substitute back into the original limit

$$
\frac{\frac{3}{x + \Delta x} - \frac{3}{x}}{\Delta x} = \frac{\frac{-3\Delta x}{x(x + \Delta x)}}{\Delta x}
$$
$$
= \frac{-3\Delta x}{x(x + \Delta x) \Delta x}
$$
$$
= \frac{-3}{x(x + \Delta x)}
$$

### Step 3: Evaluate the limit as $\Delta x \to 0^+$

$$
\lim_{\Delta x \to 0^+} \frac{-3}{x(x + \Delta x)} = \frac{-3}{x^2}
$$

**Final Answer:**
$$
\boxed{\frac{-3}{x^2}}
$$

✨ VIPSolutions

Find the limit (if it exists). (If an answer does not exist, enter DNE.) 3 E15 lim x+Ax x Ax—0* AZ

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