Sketch graphs showing vertical & horizontal asymptotes and relative extrema

• Glissando
In summary, the conversation discusses the process of sketching a graph for the function f(x) = (x2-1)/(x2-4), including finding vertical and horizontal asymptotes and relative extrema. The conversation also mentions using limits, zeroes, and derivatives to solve the problem. The main issue being discussed is with the vertical asymptote at -2, with one person stating that the limit as x->(-2)+ is +infinity and the limit as x->(-2)- is -infinity, while the other person mentions a discrepancy with their graphing calculator.
Glissando

Homework Statement

Sketch the graphs of the following function showing vertical and horizontal asymptotes and relative extrema:

f(x) = (x2-1)/(x2-4)

Homework Equations

Limits, zeroes, derivatives

The Attempt at a Solution

I know that I have the majority of the answers right, the only problem I'm having is with part of the vertical asymptote.

VA = positive and negative 2

The problem is with -2:

lim (-22-1)/(-2+)2-4)
x->-2+

= (4-1)/(4+-4) = +infinity

lim ((-2)2-1)/((-2-)2-4)
x->-2-

= (4-1)/(4--4) = negative infinity

But my graphing calculator shows the exact opposite ):!

Any help is appreciated! Thank you <3

The limit as x->(-2)+ is +infinity. The limit as x->(-2)- is -infinity. I'm not sure what your graphing calculator's problem is.

Dick said:
The limit as x->(-2)+ is +infinity. The limit as x->(-2)- is -infinity. I'm not sure what your graphing calculator's problem is.

...I guess I'm sketching the graph wrong then...Thank you for your help (:!

1. What is a vertical asymptote?

A vertical asymptote is a vertical line on a graph that the function approaches but does not touch. It occurs when the denominator of a function approaches zero, causing the function to become undefined.

2. How do you determine the location of a vertical asymptote on a graph?

To determine the location of a vertical asymptote, set the denominator of the function equal to zero and solve for the variable. The resulting value will be the x-coordinate of the vertical asymptote.

3. What is a horizontal asymptote?

A horizontal asymptote is a horizontal line on a graph that the function approaches but does not touch. It occurs when the degree of the numerator is less than the degree of the denominator in a rational function.

4. How do you find the equation of a horizontal asymptote?

The equation of a horizontal asymptote can be found by analyzing the degrees of the numerator and denominator in a rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

5. What are relative extrema on a graph?

Relative extrema are points on a graph where the function either reaches a maximum or minimum value. These points are also known as local extrema and occur at the highest or lowest points in a specific region of the graph.

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