# How do you find the vertical, horizontal or slant asymptotes for y=(x^2)/(2x^2-8)?

Feb 26, 2016

vertical asymptotes at x = ± 2
horizontal asymptote at $y = \frac{1}{2}$

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let denominator equal zero.

solve : 2x^2 - 8 = 0 → 2(x^2 - 4) = 0 → 2(x-2)(x+2) = 0

equations of vertical asymptotes are x = ± 2

Horizontal asymptotes occur as lim_(x→±∞) f(x) → 0

If the degree of the numerator and denominator are equal then the equation can be found by taking the ratio of leading coefficients.
Here they are both of degree 2.

$\Rightarrow y = \frac{1}{2}$

Here is the graph of the function.
graph{x^2/(2x^2-8) [-10, 10, -5, 5]}
The text passage provides a question about finding the vertical, horizontal, or slant asymptotes for a given function, and it also provides the answers to the question.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# How do you find the vertical, horizontal or slant asymptotes for y=(x^2)/(2x^2-8)?

Feb 26, 2016

vertical asymptotes at x = ± 2
horizontal asymptote at $y = \frac{1}{2}$

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let denominator equal zero.

solve : 2x^2 - 8 = 0 → 2(x^2 - 4) = 0 → 2(x-2)(x+2) = 0

equations of vertical asymptotes are x = ± 2

Horizontal asymptotes occur as lim_(x→±∞) f(x) → 0

If the degree of the numerator and denominator are equal then the equation can be found by taking the ratio of leading coefficients.
Here they are both of degree 2.

$\Rightarrow y = \frac{1}{2}$

Here is the graph of the function.
graph{x^2/(2x^2-8) [-10, 10, -5, 5]}
user
The text passage provides a question about finding the vertical, horizontal, or slant asymptotes for a given function, and it also provides the answers to the question.

TYPE: SINGLE_QUESTION_WITH_ANSWER