Calculus Examples

Evaluate the Limit limit as x approaches 0 of tan(9x)^x
Step 1
Use the properties of logarithms to simplify the limit.
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Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
Step 2
Set up the limit as a left-sided limit.
Step 3
Evaluate the limits by plugging in the value for the variable.
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Step 3.1
Evaluate the limit of by plugging in for .
Step 3.2
Multiply by .
Step 3.3
The exact value of is .
Step 3.4
Since is undefined, the limit does not exist.
Step 4
Set up the limit as a right-sided limit.
Step 5
Evaluate the right-sided limit.
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Step 5.1
Move the limit into the exponent.
Step 5.2
Rewrite as .
Step 5.3
Apply L'Hospital's rule.
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Step 5.3.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 5.3.1.1
Take the limit of the numerator and the limit of the denominator.
Step 5.3.1.2
As approaches from the right side, decreases without bound.
Step 5.3.1.3
Since the numerator is a constant and the denominator approaches when approaches from the right, the fraction approaches infinity.
Step 5.3.1.4
Infinity divided by infinity is undefined.
Undefined
Step 5.3.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 5.3.3
Find the derivative of the numerator and denominator.
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Step 5.3.3.1
Differentiate the numerator and denominator.
Step 5.3.3.2
Differentiate using the chain rule, which states that is where and .
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Step 5.3.3.2.1
To apply the Chain Rule, set as .
Step 5.3.3.2.2
The derivative of with respect to is .
Step 5.3.3.2.3
Replace all occurrences of with .
Step 5.3.3.3
Rewrite in terms of sines and cosines.
Step 5.3.3.4
Multiply by the reciprocal of the fraction to divide by .
Step 5.3.3.5
Write as a fraction with denominator .
Step 5.3.3.6
Simplify.
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Step 5.3.3.6.1
Rewrite the expression.
Step 5.3.3.6.2
Multiply by .
Step 5.3.3.7
Differentiate using the chain rule, which states that is where and .
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Step 5.3.3.7.1
To apply the Chain Rule, set as .
Step 5.3.3.7.2
The derivative of with respect to is .
Step 5.3.3.7.3
Replace all occurrences of with .
Step 5.3.3.8
Combine and .
Step 5.3.3.9
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.3.10
Combine and .
Step 5.3.3.11
Differentiate using the Power Rule which states that is where .
Step 5.3.3.12
Multiply by .
Step 5.3.3.13
Simplify.
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Step 5.3.3.13.1
Simplify the numerator.
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Step 5.3.3.13.1.1
Rewrite in terms of sines and cosines.
Step 5.3.3.13.1.2
Apply the product rule to .
Step 5.3.3.13.1.3
One to any power is one.
Step 5.3.3.13.1.4
Combine and .
Step 5.3.3.13.1.5
Cancel the common factor of .
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Step 5.3.3.13.1.5.1
Factor out of .
Step 5.3.3.13.1.5.2
Cancel the common factor.
Step 5.3.3.13.1.5.3
Rewrite the expression.
Step 5.3.3.13.2
Combine terms.
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Step 5.3.3.13.2.1
Rewrite as a product.
Step 5.3.3.13.2.2
Multiply by .
Step 5.3.3.14
Rewrite as .
Step 5.3.3.15
Differentiate using the Power Rule which states that is where .
Step 5.3.3.16
Rewrite the expression using the negative exponent rule .
Step 5.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.5
Combine and .
Step 5.4
Evaluate the limit.
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Step 5.4.1
Move the term outside of the limit because it is constant with respect to .
Step 5.4.2
Move the term outside of the limit because it is constant with respect to .
Step 5.5
Apply L'Hospital's rule.
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Step 5.5.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 5.5.1.1
Take the limit of the numerator and the limit of the denominator.
Step 5.5.1.2
Evaluate the limit of the numerator.
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Step 5.5.1.2.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 5.5.1.2.2
Evaluate the limit of by plugging in for .
Step 5.5.1.2.3
Raising to any positive power yields .
Step 5.5.1.3
Evaluate the limit of the denominator.
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Step 5.5.1.3.1
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 5.5.1.3.2
Move the limit inside the trig function because cosine is continuous.
Step 5.5.1.3.3
Move the term outside of the limit because it is constant with respect to .
Step 5.5.1.3.4
Move the limit inside the trig function because sine is continuous.
Step 5.5.1.3.5
Move the term outside of the limit because it is constant with respect to .
Step 5.5.1.3.6
Evaluate the limits by plugging in for all occurrences of .
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Step 5.5.1.3.6.1
Evaluate the limit of by plugging in for .
Step 5.5.1.3.6.2
Evaluate the limit of by plugging in for .
Step 5.5.1.3.7
Simplify the answer.
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Step 5.5.1.3.7.1
Multiply by .
Step 5.5.1.3.7.2
The exact value of is .
Step 5.5.1.3.7.3
Multiply by .
Step 5.5.1.3.7.4
Multiply by .
Step 5.5.1.3.7.5
The exact value of is .
Step 5.5.1.3.7.6
The expression contains a division by . The expression is undefined.
Undefined
Step 5.5.1.3.8
The expression contains a division by . The expression is undefined.
Undefined
Step 5.5.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.5.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 5.5.3
Find the derivative of the numerator and denominator.
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Step 5.5.3.1
Differentiate the numerator and denominator.
Step 5.5.3.2
Differentiate using the Power Rule which states that is where .
Step 5.5.3.3
Differentiate using the Product Rule which states that is where and .
Step 5.5.3.4
Differentiate using the chain rule, which states that is where and .
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Step 5.5.3.4.1
To apply the Chain Rule, set as .
Step 5.5.3.4.2
The derivative of with respect to is .
Step 5.5.3.4.3
Replace all occurrences of with .
Step 5.5.3.5
Raise to the power of .
Step 5.5.3.6
Raise to the power of .
Step 5.5.3.7
Use the power rule to combine exponents.
Step 5.5.3.8
Add and .
Step 5.5.3.9
Since is constant with respect to , the derivative of with respect to is .
Step 5.5.3.10
Differentiate using the Power Rule which states that is where .
Step 5.5.3.11
Multiply by .
Step 5.5.3.12
Move to the left of .
Step 5.5.3.13
Differentiate using the chain rule, which states that is where and .
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Step 5.5.3.13.1
To apply the Chain Rule, set as .
Step 5.5.3.13.2
The derivative of with respect to is .
Step 5.5.3.13.3
Replace all occurrences of with .
Step 5.5.3.14
Raise to the power of .
Step 5.5.3.15
Raise to the power of .
Step 5.5.3.16
Use the power rule to combine exponents.
Step 5.5.3.17
Add and .
Step 5.5.3.18
Since is constant with respect to , the derivative of with respect to is .
Step 5.5.3.19
Multiply by .
Step 5.5.3.20
Differentiate using the Power Rule which states that is where .
Step 5.5.3.21
Multiply by .
Step 5.6
Evaluate the limit.
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Step 5.6.1
Move the term outside of the limit because it is constant with respect to .
Step 5.6.2
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 5.6.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 5.6.4
Move the term outside of the limit because it is constant with respect to .
Step 5.6.5
Move the exponent from outside the limit using the Limits Power Rule.
Step 5.6.6
Move the limit inside the trig function because cosine is continuous.
Step 5.6.7
Move the term outside of the limit because it is constant with respect to .
Step 5.6.8
Move the term outside of the limit because it is constant with respect to .
Step 5.6.9
Move the exponent from outside the limit using the Limits Power Rule.
Step 5.6.10
Move the limit inside the trig function because sine is continuous.
Step 5.6.11
Move the term outside of the limit because it is constant with respect to .
Step 5.7
Evaluate the limits by plugging in for all occurrences of .
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Step 5.7.1
Evaluate the limit of by plugging in for .
Step 5.7.2
Evaluate the limit of by plugging in for .
Step 5.7.3
Evaluate the limit of by plugging in for .
Step 5.8
Simplify the answer.
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Step 5.8.1
Multiply by .
Step 5.8.2
Cancel the common factor of and .
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Step 5.8.2.1
Factor out of .
Step 5.8.2.2
Cancel the common factors.
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Step 5.8.2.2.1
Factor out of .
Step 5.8.2.2.2
Factor out of .
Step 5.8.2.2.3
Factor out of .
Step 5.8.2.2.4
Cancel the common factor.
Step 5.8.2.2.5
Rewrite the expression.
Step 5.8.3
Simplify the denominator.
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Step 5.8.3.1
Multiply by .
Step 5.8.3.2
The exact value of is .
Step 5.8.3.3
One to any power is one.
Step 5.8.3.4
Multiply by .
Step 5.8.3.5
The exact value of is .
Step 5.8.3.6
Raising to any positive power yields .
Step 5.8.3.7
Multiply by .
Step 5.8.3.8
Add and .
Step 5.8.4
Divide by .
Step 5.8.5
Multiply by .
Step 5.9
Anything raised to is .
Step 6
If either of the one-sided limits does not exist, the limit does not exist.