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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![The image contains a mathematical limit expression. Here is the transcription suitable for an educational website:
---
**Mathematical Limit:**
\[
\lim_{{x \to -5}} \frac{{x^2 - 25}}{{x + 5}}
\]
**Explanation:**
This limit expression requires evaluating the limit as \( x \) approaches -5 for the function \( \frac{{x^2 - 25}}{{x + 5}} \). To simplify this function, recognize that \( x^2 - 25 \) can be factored into \( (x - 5)(x + 5) \).
Substituting back into the limit expression, we get:
\[
\lim_{{x \to -5}} \frac{{(x - 5)(x + 5)}}{{x + 5}}
\]
We can cancel the common factor of \( x + 5 \) in the numerator and the denominator:
\[
\lim_{{x \to -5}} (x - 5)
\]
Now, substituting \( x = -5 \):
\[
-5 - 5 = -10
\]
Hence, the limit is:
\[
-10
\]
Understanding this process helps in comprehending the behavior of functions involving polynomials and their limits.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c88b4b6-4152-4350-b43c-eadd21488152%2Fb9de498c-0b72-4b84-bb87-30a92c9ab199%2F160ukr3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical limit expression. Here is the transcription suitable for an educational website:
---
**Mathematical Limit:**
\[
\lim_{{x \to -5}} \frac{{x^2 - 25}}{{x + 5}}
\]
**Explanation:**
This limit expression requires evaluating the limit as \( x \) approaches -5 for the function \( \frac{{x^2 - 25}}{{x + 5}} \). To simplify this function, recognize that \( x^2 - 25 \) can be factored into \( (x - 5)(x + 5) \).
Substituting back into the limit expression, we get:
\[
\lim_{{x \to -5}} \frac{{(x - 5)(x + 5)}}{{x + 5}}
\]
We can cancel the common factor of \( x + 5 \) in the numerator and the denominator:
\[
\lim_{{x \to -5}} (x - 5)
\]
Now, substituting \( x = -5 \):
\[
-5 - 5 = -10
\]
Hence, the limit is:
\[
-10
\]
Understanding this process helps in comprehending the behavior of functions involving polynomials and their limits.
---
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