Apply L'Hôpital's rule.

lim *x*→2 

*x* + 47

 − 7

*x* − 2

 = 

lim *x*→2 

*d*

*dx*

*x* + 47

 −  

*d*

*dx*

(*x* −  )

 = 

lim *x*→2 

1
 
*x* + 47

(  )

 = 

1

lim *x*→2 

1

*x* + 47

Let's apply L'Hôpital's Rule to evaluate the limit. 

Given limit:

\[
\lim_{{x \to 2}} \frac{x + 47 - 7}{x - 2}
\]

First, simplify the expression:

\[
\lim_{{x \to 2}} \frac{x + 40}{x - 2}
\]

As we discussed earlier, directly substituting \( x = 2 \) gives a division by zero, indicating that L'Hôpital's Rule may be applied. L'Hôpital's Rule states that if a limit results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), we can evaluate the limit of the derivatives of the numerator and denominator.

Differentiate the numerator and the denominator separately with respect to \(x\):

1. The derivative of the numerator \( x + 40 \) is \( 1 \).
2. The derivative of the denominator \( x - 2 \) is also \( 1 \).

Now, apply L'Hôpital's Rule:

\[
\lim_{{x \to 2}} \frac{d/dx \, (x + 40)}{d/dx \, (x - 2)} = \lim_{{x \to 2}} \frac{1}{1} = 1
\]

Thus, the limit is:

\[
\lim_{{x \to 2}} \frac{x + 40}{x - 2} = 1
\]

Would you like more details or have any questions?

---

Here are 5 related questions you might find helpful:
1. When should L'Hôpital's Rule be applied?
2. Can L'Hôpital's Rule be applied to limits involving \(\infty\)?
3. What are other methods to evaluate limits besides L'Hôpital's Rule?
4. How does L'Hôpital's Rule apply to higher-order derivatives?
5. What are common pitfalls when using L'Hôpital's Rule?

**Tip:** Always check that the limit is in an indeterminate form before applying L'Hôpital's Rule. If it is not, L'Hôpital's Rule cannot be applied correctly.

Learn how to apply L'Hôpital's Rule to evaluate the limit lim x → 2 (x + 47 - 7) / (x - 2). Understand the step-by-step process of differentiating the numerator and denominator to solve indeterminate forms. Suitable for advanced high school students and college-level calculus learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation