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Find
lim_(x rarr-3)(-5)/((x+3)^(2)).
Choose 1 answer:
(A) 0
(B)
-(5)/(36)
(c)
-(5)/(6)
(D) The limit doesn't exist

Find $\lim _{x \rightarrow-3} \frac{-5}{(x+3)^{2}}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $-\frac{5}{36}$ $\newline$ (c) $-\frac{5}{6}$ $\newline$ (D) The limit doesn't exist

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View step-by-step help

Evaluate an exponential function

Full solution

Q. Find

\lim _{x \rightarrow-3} \frac{-5}{(x+3)^{2}}

.

\newline

Choose

1

answer:

\newline

(A)

0

\newline

(B)

-\frac{5}{36}

\newline

(c)

-\frac{5}{6}

\newline

(D) The limit doesn't exist

Problem Analysis: We are asked to find the limit of the function $-\frac{5}{(x+3)^{2}}$ as $x$ approaches $-3$ . The function is a rational function, and we need to determine if the limit exists as $x$ approaches $-3$ .
Denominator Behavior: First, let's analyze the function as $x$ approaches $-3$ . The denominator $(x+3)^{2}$ becomes $0$ when $x = -3$ , which would make the function undefined at $x = -3$ . However, we are interested in the limit as $x$ approaches $-3$ , not the value of the function at $x = -3$ .
Numerator Behavior: Since the denominator $(x+3)^{2}$ is squared, it will always be positive for all $x$ except at $x = -3$ where it is $0$ . As $x$ approaches $-3$ from either side, the denominator approaches $0$ , but it does so in a positive manner because of the square.
Combining Numerator and Denominator: The numerator of the function is $-5$ , which is a constant. As $x$ approaches $-3$ , the numerator remains unchanged.
Limit Evaluation: Combining the behavior of the numerator and the denominator, as $x$ approaches $-3$ , the denominator approaches $0$ positively, and the numerator stays at $-5$ . This means the function's value will approach negative infinity because a negative number divided by a positive number that is getting closer and closer to $0$ will result in a value that becomes more and more negative.
Conclusion: Therefore, the limit of $\frac{-5}{(x+3)^{2}}$ as $x$ approaches $-3$ does not exist because the function approaches negative infinity. The correct answer is (D) The limit doesn't exist.

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