Calculus Examples

Evaluate the Limit limit as h approaches 0 of ((1-cos(h))^2)/h
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.1.2
Evaluate the limit of the numerator.
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Step 1.1.2.1
Evaluate the limit.
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Step 1.1.2.1.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.2.1.2
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.1.3
Evaluate the limit of which is constant as approaches .
Step 1.1.2.1.4
Move the limit inside the trig function because cosine is continuous.
Step 1.1.2.2
Evaluate the limit of by plugging in for .
Step 1.1.2.3
Simplify the answer.
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Step 1.1.2.3.1
Simplify each term.
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Step 1.1.2.3.1.1
The exact value of is .
Step 1.1.2.3.1.2
Multiply by .
Step 1.1.2.3.2
Subtract from .
Step 1.1.2.3.3
Raising to any positive power yields .
Step 1.1.3
Evaluate the limit of by plugging in for .
Step 1.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.2
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 1.3
Find the derivative of the numerator and denominator.
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Step 1.3.1
Differentiate the numerator and denominator.
Step 1.3.2
Differentiate using the chain rule, which states that is where and .
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Step 1.3.2.1
To apply the Chain Rule, set as .
Step 1.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3.2.3
Replace all occurrences of with .
Step 1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.5
Add and .
Step 1.3.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.7
Multiply by .
Step 1.3.8
The derivative of with respect to is .
Step 1.3.9
Multiply by .
Step 1.3.10
Simplify.
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Step 1.3.10.1
Apply the distributive property.
Step 1.3.10.2
Apply the distributive property.
Step 1.3.10.3
Combine terms.
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Step 1.3.10.3.1
Multiply by .
Step 1.3.10.3.2
Multiply by .
Step 1.3.10.4
Reorder terms.
Step 1.3.11
Differentiate using the Power Rule which states that is where .
Step 1.4
Divide by .
Step 2
Evaluate the limit.
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Step 2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.2
Move the term outside of the limit because it is constant with respect to .
Step 2.3
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 2.4
Move the limit inside the trig function because cosine is continuous.
Step 2.5
Move the limit inside the trig function because sine is continuous.
Step 2.6
Move the term outside of the limit because it is constant with respect to .
Step 2.7
Move the limit inside the trig function because sine is continuous.
Step 3
Evaluate the limits by plugging in for all occurrences of .
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Step 3.1
Evaluate the limit of by plugging in for .
Step 3.2
Evaluate the limit of by plugging in for .
Step 3.3
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Simplify each term.
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Step 4.1.1
The exact value of is .
Step 4.1.2
Multiply by .
Step 4.1.3
The exact value of is .
Step 4.1.4
Multiply by .
Step 4.1.5
The exact value of is .
Step 4.1.6
Multiply by .
Step 4.2
Add and .