Find the limits in a), b), and c) beiow for the function fix) . Use and co when appropniate.
a) Select the correct choice below and fill in any answer hoxes in your choice.
WA.
(Simplify your answer.)
B. The limit does not exst and is neither - nor .
b) Select the correct choice below and fill in any answer boxes in your choice
A. lim

(Simplify your answer.)
B. The limit does not exist and is neither nor .

Ask by Harmon Schofield. in the United States

Jan 22,2025

Question

Find the limits in a), b), and c) beiow for the function fix) . Use and co when appropniate.
a) Select the correct choice below and fill in any answer hoxes in your choice.
WA.
(Simplify your answer.)
B. The limit does not exst and is neither - nor .
b) Select the correct choice below and fill in any answer boxes in your choice
A. lim

(Simplify your answer.)
B. The limit does not exist and is neither nor .

Ask by Harmon Schofield. 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 and right.

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

**Explanation:**

- **Numerator:** As $$ x $$ approaches 7, $$ 4x $$ approaches $$ 28 $$.
- **Denominator:** As $$ x $$ approaches 7 from the left ($$ x \rightarrow 7^{-} $$), $$ x - 7 $$ approaches 0 from the **negative** side (since $$ x < 7 $$).

Thus,
$$
\lim_{x \rightarrow 7^{-}} \frac{4x}{x - 7} = \frac{28}{\text{a very small negative number}} = -\infty
$$

**Correct Choice:**

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

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

**Explanation:**

- **Numerator:** As $$ x $$ approaches 7, $$ 4x $$ approaches $$ 28 $$.
- **Denominator:** As $$ x $$ approaches 7 from the right ($$ x \rightarrow 7^{+} $$), $$ x - 7 $$ approaches 0 from the **positive** side (since $$ x > 7 $$).

Thus,
$$
\lim_{x \rightarrow 7^{+}} \frac{4x}{x - 7} = \frac{28}{\text{a very small positive number}} = +\infty
$$

**Correct Choice:**

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

---

### Summary of Answers:

- **a)** $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$
- **b)** $$ \lim_{x \rightarrow 7^{+}} f(x) = +\infty $$

If there's a part **c)**, please provide the details so I can assist you further.
a) The limit as $$ x $$ approaches 7 from the left is $$ -\infty $$. b) The limit as $$ x $$ approaches 7 from the right is $$ +\infty $$.

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