Math / Calculus

Questionp/student/3440461/25280000/44981e4a638756ec9b307cc86ca08ef3
L'Hopital's Rule Score: 0/1 Penalty: 0.25 off
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Evaluate $\lim _{x \rightarrow 3} \frac{4 \cos (3 x-9)+2 x}{5 x^{2}+5}$ and write your answer in simplest form.
Answer Attempt 1 out of 2 $\square$ Submit Answer DNE

Studdy Solution

STEP 1

1. We are given the limit $\lim _{x \rightarrow 3} \frac{4 \cos (3x-9)+2x}{5x^{2}+5}$ .
2. We need to evaluate this limit using L'Hopital's Rule if applicable.
3. L'Hopital's Rule can be applied when the limit results in an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .

STEP 2

1. Check if the limit results in an indeterminate form at $x = 3$ .
2. Apply L'Hopital's Rule if necessary.
3. Differentiate the numerator and the denominator.
4. Evaluate the new limit after differentiation.
5. Simplify the result to find the answer.

STEP 3

Substitute $x = 3$ into the original function to check for an indeterminate form:
Numerator: $4 \cos(3 \times 3 - 9) + 2 \times 3 = 4 \cos(0) + 6 = 4 \times 1 + 6 = 10$
Denominator: $5 \times 3^2 + 5 = 5 \times 9 + 5 = 45 + 5 = 50$
Since the limit does not result in an indeterminate form, L'Hopital's Rule is not necessary.

STEP 4

Since the limit does not result in an indeterminate form, evaluate the limit directly:
$\lim _{x \rightarrow 3} \frac{4 \cos (3x-9)+2x}{5x^{2}+5} = \frac{10}{50} = \frac{1}{5}$
The value of the limit is:
$\boxed{\frac{1}{5}}$

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