Find each limit for the function $$ f(x)=\frac{x^{2}-2 x-15}{x-5} $$. Use $$ -\infty $$ and $$ \infty $$ when appropriale. (A) $$ \lim _{x \rightarrow 5^{-}} f(x) $$ (B) $$ \lim _{x \rightarrow 5^{+}} f(x) $$ (C) $$ \lim _{x \rightarrow 5} f(x) $$ (A) Select the correct choice and, if necessary, till in the answer box in your choice. A. $$ \lim _{x \rightarrow 5^{-}} f(x)=\square $$ (Simplify your answer.) B. The limit does not exist and is neither - $$ \infty $$ nor $$ \infty $$.

Question

UpStudy AI · Accepted Answer

Let's analyze the function $$ f(x) = \frac{x^{2} - 2x - 15}{x - 5} $$ and compute the specified limits.

### Simplifying the Function
First, factor the numerator:
$$
x^{2} - 2x - 15 = (x - 5)(x + 3)
$$
So the function becomes:
$$
f(x) = \frac{(x - 5)(x + 3)}{x - 5}
$$
For all $$ x \neq 5 $$, we can cancel out the $$ (x - 5) $$ terms:
$$
f(x) = x + 3 \quad \text{for} \quad x \neq 5
$$

### (A) $$ \lim_{x \to 5^{-}} f(x) $$
As $$ x $$ approaches $$ 5 $$ from the left ($$ x \to 5^- $$):
$$
f(x) = x + 3 \quad \Rightarrow \quad \lim_{x \to 5^{-}} f(x) = 5 + 3 = 8
$$

### (B) $$ \lim_{x \to 5^{+}} f(x) $$
As $$ x $$ approaches $$ 5 $$ from the right ($$ x \to 5^+ $$):
$$
f(x) = x + 3 \quad \Rightarrow \quad \lim_{x \to 5^{+}} f(x) = 5 + 3 = 8
$$

### (C) $$ \lim_{x \to 5} f(x) $$
Since both the left-hand and right-hand limits are equal:
$$
\lim_{x \to 5} f(x) = 8
$$

### Summary of Limits
- **(A)** $$ \lim_{x \to 5^{-}} f(x) = 8 $$
- **(B)** $$ \lim_{x \to 5^{+}} f(x) = 8 $$
- **(C)** $$ \lim_{x \to 5} f(x) = 8 $$

All three limits exist and are equal to $$ 8 $$.
(A) $$ \lim_{x \to 5^{-}} f(x) = 8 $$ (B) $$ \lim_{x \to 5^{+}} f(x) = 8 $$ (C) $$ \lim_{x \to 5} f(x) = 8 $$

Find each limit for the function . Use and when appropriale.
(A)
(B)
©
(A) Select the correct choice and, if necessary, till in the answer box in your choice.
A. (Simplify your answer.)
B. The limit does not exist and is neither - nor .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

Find each limit for the function . Use and when appropriale. (A) (B) © (A) Select the correct choice and, if necessary, till in the answer box in your choice. A. (Simplify your answer.) B. The limit does not exist and is neither - nor .