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Consider f(x) = x² - 16 X 4 X 3.9 3.99 3.999 3.9999 f(x) 7.9 X 4.1 4.01 4.001 4.0001 f(x) 8.1 Looking at the values, we can estimate that lim f(x) = = x →4

Consider f(x) = x² - 16 X 4 X 3.9 3.99 3.999 3.9999 f(x) 7.9 X 4.1 4.01 4.001 4.0001 f(x) 8.1 Looking at the values, we can estimate that lim f(x) = = x →4

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Consider the Function \( f(x) = \frac{x^2 - 16}{x - 4} \)** This expression involves a rational function. Let's examine how \( f(x) \) behaves as \( x \) approaches 4. **Table Analysis** The table displays values of \( x \) and corresponding values of \( f(x) \) as \( x \) approaches 4 from both sides (values less than 4 and more than 4). - **For x approaching 4 from the left (values less than 4):** - \( x = 3.9 \), \( f(x) = 7.9 \) - \( x = 3.99 \), \( f(x) = \) [Not provided] - \( x = 3.999 \), \( f(x) = \) [Not provided] - \( x = 3.9999 \), \( f(x) = \) [Not provided] - **For x approaching 4 from the right (values greater than 4):** - \( x = 4.1 \), \( f(x) = 8.1 \) - \( x = 4.01 \), \( f(x) = \) [Not provided] - \( x = 4.001 \), \( f(x) = \) [Not provided] - \( x = 4.0001 \), \( f(x) = \) [Not provided] **Estimating the Limit** As we examine these values, it suggests the behavior of the function near the point \( x = 4 \). Observing both the left and right approaches will help us determine: \[ \lim_{x \to 4} f(x) = \] The information encourages us to fill out the missing \( f(x) \) values and establish the limit based on visible trends.
Transcribed Image Text:**Consider the Function \( f(x) = \frac{x^2 - 16}{x - 4} \)** This expression involves a rational function. Let's examine how \( f(x) \) behaves as \( x \) approaches 4. **Table Analysis** The table displays values of \( x \) and corresponding values of \( f(x) \) as \( x \) approaches 4 from both sides (values less than 4 and more than 4). - **For x approaching 4 from the left (values less than 4):** - \( x = 3.9 \), \( f(x) = 7.9 \) - \( x = 3.99 \), \( f(x) = \) [Not provided] - \( x = 3.999 \), \( f(x) = \) [Not provided] - \( x = 3.9999 \), \( f(x) = \) [Not provided] - **For x approaching 4 from the right (values greater than 4):** - \( x = 4.1 \), \( f(x) = 8.1 \) - \( x = 4.01 \), \( f(x) = \) [Not provided] - \( x = 4.001 \), \( f(x) = \) [Not provided] - \( x = 4.0001 \), \( f(x) = \) [Not provided] **Estimating the Limit** As we examine these values, it suggests the behavior of the function near the point \( x = 4 \). Observing both the left and right approaches will help us determine: \[ \lim_{x \to 4} f(x) = \] The information encourages us to fill out the missing \( f(x) \) values and establish the limit based on visible trends.
Let \( f(x) = \begin{cases} 4 - x - x^2 & \text{if } x \leq 4 \\ 2x - 24 & \text{if } x > 4 \end{cases} \) Calculate the following limits. Enter "DNE" if the limit does not exist. \[ \lim_{x \to 4^-} f(x) = \underline{\hspace{2cm}} \] \[ \lim_{x \to 4^+} f(x) = \underline{\hspace{2cm}} \] \[ \lim_{x \to 4} f(x) = \underline{\hspace{2cm}} \]
Transcribed Image Text:Let \( f(x) = \begin{cases} 4 - x - x^2 & \text{if } x \leq 4 \\ 2x - 24 & \text{if } x > 4 \end{cases} \) Calculate the following limits. Enter "DNE" if the limit does not exist. \[ \lim_{x \to 4^-} f(x) = \underline{\hspace{2cm}} \] \[ \lim_{x \to 4^+} f(x) = \underline{\hspace{2cm}} \] \[ \lim_{x \to 4} f(x) = \underline{\hspace{2cm}} \]
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