Calculus Examples

Evaluate the Limit limit as x approaches 0 of (sin(x)^2)/x
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
Move the limit inside the trig function because sine 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
The exact value of is .
Step 1.1.2.3.2
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
The derivative of with respect to is .
Step 1.3.4
Simplify.
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Step 1.3.4.1
Reorder the factors of .
Step 1.3.4.2
Reorder and .
Step 1.3.4.3
Reorder and .
Step 1.3.4.4
Apply the sine double-angle identity.
Step 1.3.5
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
Move the limit inside the trig function because sine is continuous.
Step 2.2
Move the term outside of the limit because it is constant with respect to .
Step 3
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Multiply by .
Step 4.2
The exact value of is .