Find the limits in a), b), and c) be ow for the function . Use and when appropriate.
a) Select the correct choice below and fill in any answer boxes in your choice.
A.

(S mplify your answer.)
B. The limit does not exist and is neither nor .
b) Select the correct choice balow and fill in any answer boxes in your choice
A.

(Simplify your answer:)
B. The limit does not exist and is neither nor .
c) Select the correct choice below and fill in any answer boxes in your choice.
A.
(Simplify your answer)
B. The limit does not exist and is neither nor .

Ask by Matthews Huff. in the United States

Jan 22,2025

Question

Find the limits in a), b), and c) be ow for the function . Use and when appropriate.
a) Select the correct choice below and fill in any answer boxes in your choice.
A.

(S mplify your answer.)
B. The limit does not exist and is neither nor .
b) Select the correct choice balow and fill in any answer boxes in your choice
A.

(Simplify your answer:)
B. The limit does not exist and is neither nor .
c) Select the correct choice below and fill in any answer boxes in your choice.
A.
(Simplify your answer)
B. The limit does not exist and is neither nor .

Ask by Matthews Huff. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

Let's analyze the limits of the function $$ f(x) = \frac{4x}{x - 7} $$ as $$ x $$ approaches 7 from the left ($$ 7^- $$), from the right ($$ 7^+ $$), and overall ($$ 7 $$).

### a) $$ \lim_{x \rightarrow 7^{-}} f(x) $$

**Analysis:**
- As $$ x $$ approaches 7 from the left ($$ x \rightarrow 7^{-} $$):
  - The denominator $$ x - 7 $$ approaches 0 from the negative side.
  - The numerator $$ 4x $$ approaches $$ 28 $$.
  
Therefore:
$$
\lim_{x \rightarrow 7^{-}} \frac{4x}{x - 7} = \frac{28}{0^-} = -\infty
$$

**Choice:**
- **A.**
  $$
  \lim_{x \rightarrow 7^{-}} f(x) = -\infty
  $$

---

### b) $$ \lim_{x \rightarrow 7^{+}} f(x) $$

**Analysis:**
- As $$ x $$ approaches 7 from the right ($$ x \rightarrow 7^{+} $$):
  - The denominator $$ x - 7 $$ approaches 0 from the positive side.
  - The numerator $$ 4x $$ approaches $$ 28 $$.

Therefore:
$$
\lim_{x \rightarrow 7^{+}} \frac{4x}{x - 7} = \frac{28}{0^+} = \infty
$$

**Choice:**
- **A.**
  $$
  \lim_{x \rightarrow 7^{+}} f(x) = \infty
  $$

---

### c) $$ \lim_{x \rightarrow 7} f(x) $$

**Analysis:**
- From part **a**, $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$.
- From part **b**, $$ \lim_{x \rightarrow 7^{+}} f(x) = \infty $$.

Since the left-hand limit and the right-hand limit are not equal (one approaches $$ -\infty $$ and the other approaches $$ \infty $$), the overall limit does not exist.

**Choice:**
- **B.** The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

---

### Summary of Choices:

- **a)** **A.**
  $$
  \lim_{x \rightarrow 7^{-}} f(x) = -\infty
  $$
  
- **b)** **A.**
  $$
  \lim_{x \rightarrow 7^{+}} f(x) = \infty
  $$
  
- **c)** **B.** The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.
a) $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$ b) $$ \lim_{x \rightarrow 7^{+}} f(x) = \infty $$ c) The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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