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Trigonometry Examples
Step 1
Step 1.1
Find where the expression is undefined.
Step 1.2
Ignoring the logarithm, consider the rational function where is the degree of the numerator and is the degree of the denominator.
1. If , then the x-axis, , is the horizontal asymptote.
2. If , then the horizontal asymptote is the line .
3. If , then there is no horizontal asymptote (there is an oblique asymptote).
Step 1.3
Find and .
Step 1.4
Since , the x-axis, , is the horizontal asymptote.
Step 1.5
No oblique asymptotes are present for logarithmic and trigonometric functions.
No Oblique Asymptotes
Step 1.6
This is the set of all asymptotes.
Vertical Asymptotes:
Horizontal Asymptotes:
Vertical Asymptotes:
Horizontal Asymptotes:
Step 2
Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
Step 2.2.1
Simplify by moving inside the logarithm.
Step 2.2.2
One to any power is one.
Step 2.2.3
Simplify the numerator.
Step 2.2.3.1
One to any power is one.
Step 2.2.3.2
Logarithm base of is .
Step 2.2.4
Simplify the expression.
Step 2.2.4.1
Multiply by .
Step 2.2.4.2
Divide by .
Step 2.2.5
The final answer is .
Step 2.3
Convert to decimal.
Step 3
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Step 3.2.1
Simplify by moving inside the logarithm.
Step 3.2.2
Raise to the power of .
Step 3.2.3
Raise to the power of .
Step 3.2.4
Multiply by .
Step 3.2.5
Rewrite as .
Step 3.2.6
Expand by moving outside the logarithm.
Step 3.2.7
Cancel the common factor of and .
Step 3.2.7.1
Factor out of .
Step 3.2.7.2
Cancel the common factors.
Step 3.2.7.2.1
Factor out of .
Step 3.2.7.2.2
Cancel the common factor.
Step 3.2.7.2.3
Rewrite the expression.
Step 3.2.8
Rewrite as .
Step 3.2.9
Simplify by moving inside the logarithm.
Step 3.2.10
The final answer is .
Step 3.3
Convert to decimal.
Step 4
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify by moving inside the logarithm.
Step 4.2.2
Raise to the power of .
Step 4.2.3
Simplify the numerator.
Step 4.2.3.1
Use logarithm rules to move out of the exponent.
Step 4.2.3.2
Logarithm base of is .
Step 4.2.3.3
Multiply by .
Step 4.2.4
Multiply by .
Step 4.2.5
The final answer is .
Step 4.3
Convert to decimal.
Step 5
The log function can be graphed using the vertical asymptote at and the points .
Vertical Asymptote:
Step 6