PLEASE BE ACURRATE! HURRY ### Understanding Limits with Rational FunctionsConsider the following function:\[ f(x) = \frac{x^2}{x^2 - 16} \]### Completing the TableWe want to explore the behavior of $ f(x) $ as $ x $ approaches $-4$. Complete the following table by calculating $ f(x) $ for each given value of $ x $. Round your answers to two decimal places.| $ x $ | $-4.5$ | $-4.1$ | $-4.01$ | $-4.001$ | $-4$ | $-3.999$ | $-3.99$ | $-3.9$ | $-3.5$ ||------------|----------|----------|-----------|------------|-------|------------|-----------|----------|----------|| $ f(x) $ | | | | | ? | | | | |### Evaluating LimitsUse the table to determine whether $ f(x) $ approaches $ \infty $ or $ -\infty $ as $ x $ approaches $-4$ from the left and from the right.\[ \lim_{{x \to -4^-}} f(x) = \]\[ \lim_{{x \to -4^+}} f(x) = \]### Instructions:1. **Calculate $ f(x) $**: For each $ x $ value in the table, compute the function $ f(x) $ and fill in the corresponding cell.2. **Analyze the Limits**: Use the table values to infer the behavior of $ f(x) $ as $ x $ approaches $-4$ from the left side ($ x \to -4^- $) and from the right side ($ x \to -4^+ $).3. **Fill in Limit Values**: Based on the function's behavior near $ x = -4 $, determine if the function approaches positive infinity ($ \infty $), negative infinity ($ -\infty $), or if it is undefined.By understanding how to find the limit of a rational function through numerical substitution and pattern recognition, you can apply these methods to a variety of functions and improve your understanding of limits and continuity in mathematical

To find the limit of a function f(x) at a point , calculate the right and left hand side of limit of…

Consider the following function. x2 f(x) = x2 - 16 Complete the following table. (Round your answers to two decimal places.) -4.5 -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9 -3.5 f(x) Use the table to determine whether f(x) approaches o or -o as x approaches -4 from the left and from the right. lim f(x) = x--4 lim f(x) = x--4*

Consider the following function. x2 f(x) = x2 - 16 Complete the following table. (Round your answers to two decimal places.) -4.5 -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9 -3.5 f(x) Use the table to determine whether f(x) approaches o or -o as x approaches -4 from the left and from the right. lim f(x) = x--4 lim f(x) = x--4*

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

PLEASE BE ACURRATE! HURRY

### Understanding Limits with Rational Functions

Consider the following function:
\[ f(x) = \frac{x^2}{x^2 - 16} \]

### Completing the Table
We want to explore the behavior of $ f(x) $ as $ x $ approaches $-4$. Complete the following table by calculating $ f(x) $ for each given value of $ x $. Round your answers to two decimal places.

| $ x $ | $-4.5$ | $-4.1$ | $-4.01$ | $-4.001$ | $-4$ | $-3.999$ | $-3.99$ | $-3.9$ | $-3.5$ |
|------------|----------|----------|-----------|------------|-------|------------|-----------|----------|----------|
| $ f(x) $ | | | | | ? | | | | |

### Evaluating Limits
Use the table to determine whether $ f(x) $ approaches $ \infty $ or $ -\infty $ as $ x $ approaches $-4$ from the left and from the right.

\[ \lim_{{x \to -4^-}} f(x) = \]
\[ \lim_{{x \to -4^+}} f(x) = \]

### Instructions:
1. **Calculate $ f(x) $**: For each $ x $ value in the table, compute the function $ f(x) $ and fill in the corresponding cell.
2. **Analyze the Limits**: Use the table values to infer the behavior of $ f(x) $ as $ x $ approaches $-4$ from the left side ($ x \to -4^- $) and from the right side ($ x \to -4^+ $).
3. **Fill in Limit Values**: Based on the function's behavior near $ x = -4 $, determine if the function approaches positive infinity ($ \infty $), negative infinity ($ -\infty $), or if it is undefined.

By understanding how to find the limit of a rational function through numerical substitution and pattern recognition, you can apply these methods to a variety of functions and improve your understanding of limits and continuity in mathematical

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