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

(a) using techniques from Chapters 2 and 4
(b) using L’Hôpital’s Rule
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Ask by Franklin Carlson. 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
(b) using L’Hôpital’s Rule
Need Help? Readlit

Ask by Franklin Carlson. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

**(a) Using techniques from Chapters 2 and 4**

We start with the limit
$$
\lim_{x \to 0} \frac{\sin(7x)}{8x}.
$$
Recall the standard limit
$$
\lim_{u \to 0} \frac{\sin u}{u} = 1.
$$
Let $$ u = 7x $$, so that as $$ x \to 0 $$, $$ u \to 0 $$ and $$ \sin(7x) = \sin u $$. Notice that
$$
8x = \frac{8}{7} \cdot 7x = \frac{8}{7} \cdot u.
$$
Thus, we can rewrite the limit as
$$
\lim_{u \to 0} \frac{\sin u}{\frac{8}{7} u} = \frac{7}{8} \lim_{u \to 0} \frac{\sin u}{u}.
$$
Since
$$
\lim_{u \to 0} \frac{\sin u}{u} = 1,
$$
the limit becomes
$$
\frac{7}{8} \times 1 = \frac{7}{8}.
$$

**(b) Using L'Hôpital's Rule**

We again consider the limit
$$
\lim_{x \to 0} \frac{\sin(7x)}{8x}.
$$
Since both the numerator and denominator approach 0 as $$ x \to 0 $$, we apply L'Hôpital's Rule by differentiating the numerator and denominator with respect to $$ x $$.

Differentiate the numerator:
$$
\frac{d}{dx} \sin(7x) = 7 \cos(7x).
$$
Differentiate the denominator:
$$
\frac{d}{dx} (8x) = 8.
$$

Thus, by L'Hôpital's Rule, the limit becomes
$$
\lim_{x \to 0} \frac{7 \cos(7x)}{8}.
$$
Evaluating at $$ x = 0 $$:
$$
\frac{7 \cos(0)}{8} = \frac{7 \cdot 1}{8} = \frac{7}{8}.
$$

Therefore, both methods yield the same result:
$$
\frac{7}{8}.
$$
The limit is $$ \frac{7}{8} $$.

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

(a) using techniques from Chapters 2 and 4
(b) using L’Hôpital’s Rule
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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 (b) using L’Hôpital’s Rule Need Help? Readlit

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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
(b) using L’Hôpital’s Rule
Need Help? Readlit