give answer in 2 step with explanation at the end of each step and final answer at the end:J
evaluate the limit: lim (1+ 4x) ¥
x= Io}

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:J
evaluate the limit: lim (1+ 4x) ¥
x= Io}

VIPSolutions · Accepted Answer

## Step 1: Rewrite the Limit in Exponential Form

Given:
$$
\lim_{x \to ^+} (1 + 4x)^{\frac{1}{x}}
$$
Let’s take the natural logarithm to simplify the exponent:
Let $ y = (1 + 4x)^{1/x} $.

Taking natural log on both sides:
$$
\ln y = \frac{1}{x} \ln(1 + 4x)
$$

**Explanation:**  
Taking the logarithm brings the exponent down, making the limit easier to evaluate.

---

## Step 2: Evaluate the Limit Using L'Hôpital's Rule

Now, consider:
$$
\lim_{x \to ^+} \frac{\ln(1 + 4x)}{x}
$$
This is an indeterminate form $ \frac{}{} $, so apply L'Hôpital’s Rule:

Differentiate numerator and denominator:
$$
\lim_{x \to ^+} \frac{4}{1+4x}
$$
As $ x \to ^+ $, $ 1+4x \to 1 $, so the limit is 4.

So,
$$
\ln y = 4 \implies y = e^4
$$

**Explanation:**  
L'Hôpital’s Rule helps evaluate the indeterminate form, leading to the answer.

---

## **Final Answer**

$$
\boxed{e^4}
$$

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