### Evaluating Limits Using a Table of ValuesTo help you understand the process of evaluating limits, let's explore the function given below and create a table of values to approach the limit as $ x $ approaches $-5$.#### Function:\[ f(x) = \frac{x^2 - 2x - 35}{x + 5} \]#### Table of Values:We will calculate $ f(x) $ for values of $ x $ approaching $-5$ from both the left and the right. The chosen values are: $-5.1$, $-5.01$, $-5.001$, $-5.0001$, $-4.999$, $-4.99$, and $-4.9$.| $ x $ | -5.1 | -5.01 | -5.001 | -5.0001 | -4.999 | -4.99 | -4.9 ||------------|-----------|-----------|-----------|-----------|-----------|----------|----------|| $ f(x) $ | | | | | | | |#### Explanation of the Table:- $ x $: These are the x-values for which we will compute $ f(x) $.- $ f(x) $: These are the corresponding values of the function $ f(x) $.The goal is to evaluate the limit as $ x $ approaches $-5$:\[ \lim_{x \to -5} f(x) \]Fill in the table by substituting each $ x $-value into the function $ f(x) $. Calculate the corresponding $ f(x) $ values.By analyzing the behavior of $ f(x) $ as $ x $ gets closer to $-5$, you can determine the limit. When $ x $ is close to $ -5 $, if $ f(x) $ approaches a specific number, that number is the limit.After filling in the table and observing the trend in $ f(x) $, you will conclude:\[ \lim_{x \to -5} f(x) = \text{(value)} \]This method helps visualize how $ f(x) $ behaves near the point of interest and aids in understanding the concept of limits.

Answered: Image /qna-images/answer/ed4061b0-1606-46d3-8bdc-c3ceab8f6224.jpg

Answered: x²-2x-35 3. f(x) = x+5 -5.1 2.1 -5.001…

Math

Trigonometry

x²-2x-35 3. f(x) = x+5 -5.1 2.1 -5.001 -5 -4.999 -4.99 -4.9 х f(x) lim f(x) = %3D x--5

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

### Evaluating Limits Using a Table of Values

To help you understand the process of evaluating limits, let's explore the function given below and create a table of values to approach the limit as $ x $ approaches $-5$.

#### Function:
\[ f(x) = \frac{x^2 - 2x - 35}{x + 5} \]

#### Table of Values:
We will calculate $ f(x) $ for values of $ x $ approaching $-5$ from both the left and the right. The chosen values are: $-5.1$, $-5.01$, $-5.001$, $-5.0001$, $-4.999$, $-4.99$, and $-4.9$.

| $ x $ | -5.1 | -5.01 | -5.001 | -5.0001 | -4.999 | -4.99 | -4.9 |
|------------|-----------|-----------|-----------|-----------|-----------|----------|----------|
| $ f(x) $ | | | | | | | |

#### Explanation of the Table:
- $ x $: These are the x-values for which we will compute $ f(x) $.
- $ f(x) $: These are the corresponding values of the function $ f(x) $.

The goal is to evaluate the limit as $ x $ approaches $-5$:
\[ \lim_{x \to -5} f(x) \]

Fill in the table by substituting each $ x $-value into the function $ f(x) $. Calculate the corresponding $ f(x) $ values.

By analyzing the behavior of $ f(x) $ as $ x $ gets closer to $-5$, you can determine the limit. When $ x $ is close to $ -5 $, if $ f(x) $ approaches a specific number, that number is the limit.

After filling in the table and observing the trend in $ f(x) $, you will conclude:
\[ \lim_{x \to -5} f(x) = \text{(value)} \]

This method helps visualize how $ f(x) $ behaves near the point of interest and aids in understanding the concept of limits.

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