QuestionFor the following exercises, find the horizontal and vertical asymptotes.
271.
272.
273.
274.
275.
276.
277.
278.
279.
280.
Studdy Solution
STEP 1
1. Horizontal asymptotes occur when the limit of as approaches a constant value.
2. Vertical asymptotes occur where the function is undefined and the limit of approaches .
3. Rational functions typically have vertical asymptotes where the denominator is zero, provided the numerator is not also zero at those points.
4. Trigonometric functions can have vertical asymptotes where they are undefined.
STEP 2
1. Analyze each function for horizontal asymptotes.
2. Analyze each function for vertical asymptotes.
STEP 3
For each function, find the horizontal asymptotes by evaluating the limits as .
STEP 4
For each function, find the vertical asymptotes by determining where the function is undefined and evaluating the limits approaching those points.
STEP 5
Analyze :
- Horizontal Asymptote: As , , so there is no horizontal asymptote.
- Vertical Asymptote: None, since the function is defined for all .
STEP 6
Analyze :
- Horizontal Asymptote: As , .
- Vertical Asymptote: At , since the denominator is zero.
STEP 7
Analyze :
- Horizontal Asymptote: None, as the degree of the numerator is greater than the denominator.
- Vertical Asymptote: At , since the denominator is zero.
STEP 8
Analyze :
- Horizontal Asymptote: As , .
- Vertical Asymptote: None, as the denominator is never zero.
STEP 9
Analyze :
- Horizontal Asymptote: None, as trigonometric functions oscillate.
- Vertical Asymptote: None, as sine is defined for all .
STEP 10
Analyze :
- Horizontal Asymptote: None, as trigonometric functions oscillate.
- Vertical Asymptote: None, as cosine is defined for all .
STEP 11
Analyze :
- Horizontal Asymptote: None, as the numerator and denominator have the same degree.
- Vertical Asymptote: At , since the denominator is zero.
STEP 12
Analyze :
- Horizontal Asymptote: None, as the function is undefined at multiples of .
- Vertical Asymptote: At , where is an integer, since .
STEP 13
Analyze :
- Horizontal Asymptote: As , .
- Vertical Asymptote: At and , since the denominator is zero.
STEP 14
Analyze :
- Horizontal Asymptote: As , , so no horizontal asymptote.
- Vertical Asymptote: At , since the denominator of the first term is zero.
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