Peterson Adkins
11/22/2023 · Middle School
\( \lim _ { x \rightarrow x 5 } \sqrt { x ^ { 2 } + 4 } ( 5 x ^ { 2 } 4 ) ( 3 x + 7 ) ^ { 2 } \)
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Tutor-Verified Answer
Step-by-step Solution
To evaluate the limit
\[
\lim_{x \rightarrow 5} \sqrt{x^2 + 4} (5x^2 - 4) (3x + 7)^2,
\]
we will substitute \( x = 5 \) directly into the expression, as there are no indeterminate forms present.
1. **Calculate \( \sqrt{x^2 + 4} \) at \( x = 5 \)**:
\[
\sqrt{5^2 + 4} = \sqrt{25 + 4} = \sqrt{29}.
\]
2. **Calculate \( 5x^2 - 4 \) at \( x = 5 \)**:
\[
5(5^2) - 4 = 5(25) - 4 = 125 - 4 = 121.
\]
3. **Calculate \( (3x + 7)^2 \) at \( x = 5 \)**:
\[
(3(5) + 7)^2 = (15 + 7)^2 = 22^2 = 484.
\]
Now, we can combine these results to find the limit:
\[
\lim_{x \rightarrow 5} \sqrt{x^2 + 4} (5x^2 - 4) (3x + 7)^2 = \sqrt{29} \cdot 121 \cdot 484.
\]
Next, we compute the product:
1. **Calculate \( 121 \cdot 484 \)**:
\[
121 \cdot 484 = 58564.
\]
2. **Now multiply by \( \sqrt{29} \)**:
\[
\sqrt{29} \cdot 58564.
\]
Thus, the final result is:
\[
\lim_{x \rightarrow 5} \sqrt{x^2 + 4} (5x^2 - 4) (3x + 7)^2 = 58564 \sqrt{29}.
\]
Therefore, the limit is
\[
\boxed{58564 \sqrt{29}}.
\]
Quick Answer
The limit is \( 58564 \sqrt{29} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
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