Evaluate the limit using techniques from Chapters 2 and 4 and using L’Hôpital’s Rule.

(a) using techniques from Chapters 2 and 4
0
(b) using L’Hôpital’s Rưle

Ask by Watson Owen. in the United States

Mar 31,2025

Question

Evaluate the limit using techniques from Chapters 2 and 4 and using L’Hôpital’s Rule.

(a) using techniques from Chapters 2 and 4
0
(b) using L’Hôpital’s Rưle

Ask by Watson Owen. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

**Step 1. Factor the Denominator**

The given limit is  
$$
\lim_{x \to 9} \frac{5(x-9)}{x^2-81}.
$$  
Notice that the denominator can be factored as a difference of squares:
$$
x^2 - 81 = (x-9)(x+9).
$$

**Step 2. Simplify the Expression (Techniques from Chapters 2 and 4)**

Substitute the factored form into the limit:
$$
\lim_{x \to 9} \frac{5(x-9)}{(x-9)(x+9)}.
$$
Cancel the common factor $$(x-9)$$ (this cancellation is valid since we are considering the limit as $$x$$ approaches 9, not $$x=9$$ itself):
$$
\lim_{x \to 9} \frac{5}{x+9}.
$$
Now substitute $$x=9$$:
$$
\frac{5}{9+9} = \frac{5}{18}.
$$

**Step 3. Evaluate Using L'Hôpital's Rule**

Since both the numerator and the denominator approach 0 as $$x \to 9$$:
- Numerator: $$5(x-9) \to 5(9-9)=0$$.
- Denominator: $$x^2-81 \to 9^2-81 = 81-81=0$$.

This creates an indeterminate form $$\frac{0}{0}$$. We apply L'Hôpital's Rule by differentiating the numerator and the denominator.

Differentiate the numerator:
$$
\frac{d}{dx}[5(x-9)] = 5.
$$

Differentiate the denominator:
$$
\frac{d}{dx}[x^2-81] = 2x.
$$

Thus, the limit becomes:
$$
\lim_{x \to 9} \frac{5}{2x}.
$$
Now substitute $$x = 9$$:
$$
\frac{5}{2 \times 9} = \frac{5}{18}.
$$

**Final Answers:**

(a) Using techniques from Chapters 2 and 4: $$\frac{5}{18}$$

(b) Using L'Hôpital's Rule: $$\frac{5}{18}$$
The limit is $$\frac{5}{18}$$ in both methods.

Evaluate the limit using techniques from Chapters 2 and 4 and using L’Hôpital’s Rule.

(a) using techniques from Chapters 2 and 4
0
(b) using L’Hôpital’s Rưle

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Evaluate the limit using techniques from Chapters 2 and 4 and using L’Hôpital’s Rule. (a) using techniques from Chapters 2 and 4 0 (b) using L’Hôpital’s Rưle

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Evaluate the limit using techniques from Chapters 2 and 4 and using L’Hôpital’s Rule.

(a) using techniques from Chapters 2 and 4
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(b) using L’Hôpital’s Rưle