Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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 topic…
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"Graph the function. Show work with finding the important info, including using limits where necessary."

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### Understanding Rational Functions #### Function: \[ f(x) = -\frac{(x-4)(x+5)(x-7)}{(x-2)(x+1)} \] #### X and Y Intercepts: #### Hole(s): #### Horizontal Asymptote(s): #### Vertical Asymptote(s) and Number Line: #### Slant Asymptote: #### Graph: --- ### Explanation **X and Y Intercepts**: - **X-intercept(s)**: Points where \( f(x) = 0 \). This occurs when the numerator is equal to zero, provided the zeros are not canceled out by the denominator. - **Y-intercept**: This occurs when \( x = 0 \). **Hole(s)**: - Holes occur where a factor in the numerator and denominator are the same, indicating the graph is undefined at that point, but there is a removable discontinuity. **Horizontal Asymptote(s)**: - This is determined by the degrees of the polynomial in the numerator and denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. **Vertical Asymptote(s) and Number Line**: - Vertical asymptotes occur where the denominator is zero (provided the numerator is not zero at the same point). - Draw the number line to indicate the behavior of the function around these points. **Slant Asymptote**: - Found if the degree of the numerator is exactly one more than the degree of the denominator. Polynomial long division can be used to determine the equation of the slant asymptote. **Graph**: - A visual representation of the function including all intercepts, asymptotes, and holes. This template allows users to fill in detailed explanations and graphical elements to understand the given rational function thoroughly.
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Transcribed Image Text:### Understanding Rational Functions #### Function: \[ f(x) = -\frac{(x-4)(x+5)(x-7)}{(x-2)(x+1)} \] #### X and Y Intercepts: #### Hole(s): #### Horizontal Asymptote(s): #### Vertical Asymptote(s) and Number Line: #### Slant Asymptote: #### Graph: --- ### Explanation **X and Y Intercepts**: - **X-intercept(s)**: Points where \( f(x) = 0 \). This occurs when the numerator is equal to zero, provided the zeros are not canceled out by the denominator. - **Y-intercept**: This occurs when \( x = 0 \). **Hole(s)**: - Holes occur where a factor in the numerator and denominator are the same, indicating the graph is undefined at that point, but there is a removable discontinuity. **Horizontal Asymptote(s)**: - This is determined by the degrees of the polynomial in the numerator and denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. **Vertical Asymptote(s) and Number Line**: - Vertical asymptotes occur where the denominator is zero (provided the numerator is not zero at the same point). - Draw the number line to indicate the behavior of the function around these points. **Slant Asymptote**: - Found if the degree of the numerator is exactly one more than the degree of the denominator. Polynomial long division can be used to determine the equation of the slant asymptote. **Graph**: - A visual representation of the function including all intercepts, asymptotes, and holes. This template allows users to fill in detailed explanations and graphical elements to understand the given rational function thoroughly.
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