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Calculus Examples
Step 1
Multiply the numerator and denominator by .
Step 2
Multiply the numerator and denominator by .
Step 3
Separate fractions.
Step 4
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 5
Step 5.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 5.1.1
Take the limit of the numerator and the limit of the denominator.
Step 5.1.2
Evaluate the limit of the numerator.
Step 5.1.2.1
Evaluate the limit.
Step 5.1.2.1.1
Move the limit inside the trig function because sine is continuous.
Step 5.1.2.1.2
Move the term outside of the limit because it is constant with respect to .
Step 5.1.2.2
Evaluate the limit of by plugging in for .
Step 5.1.2.3
Simplify the answer.
Step 5.1.2.3.1
Multiply by .
Step 5.1.2.3.2
The exact value of is .
Step 5.1.3
Evaluate the limit of the denominator.
Step 5.1.3.1
Move the term outside of the limit because it is constant with respect to .
Step 5.1.3.2
Evaluate the limit of by plugging in for .
Step 5.1.3.3
Multiply by .
Step 5.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 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 5.3
Find the derivative of the numerator and denominator.
Step 5.3.1
Differentiate the numerator and denominator.
Step 5.3.2
Differentiate using the chain rule, which states that is where and .
Step 5.3.2.1
To apply the Chain Rule, set as .
Step 5.3.2.2
The derivative of with respect to is .
Step 5.3.2.3
Replace all occurrences of with .
Step 5.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.4
Differentiate using the Power Rule which states that is where .
Step 5.3.5
Multiply by .
Step 5.3.6
Move to the left of .
Step 5.3.7
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.8
Differentiate using the Power Rule which states that is where .
Step 5.3.9
Multiply by .
Step 5.4
Evaluate the limit.
Step 5.4.1
Cancel the common factor of .
Step 5.4.1.1
Cancel the common factor.
Step 5.4.1.2
Divide by .
Step 5.4.2
Move the limit inside the trig function because cosine is continuous.
Step 5.4.3
Move the term outside of the limit because it is constant with respect to .
Step 5.5
Evaluate the limit of by plugging in for .
Step 5.6
Simplify the answer.
Step 5.6.1
Multiply by .
Step 5.6.2
The exact value of is .
Step 6
Step 6.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 6.1.1
Take the limit of the numerator and the limit of the denominator.
Step 6.1.2
Evaluate the limit of the numerator.
Step 6.1.2.1
Move the term outside of the limit because it is constant with respect to .
Step 6.1.2.2
Evaluate the limit of by plugging in for .
Step 6.1.2.3
Multiply by .
Step 6.1.3
Evaluate the limit of the denominator.
Step 6.1.3.1
Evaluate the limit.
Step 6.1.3.1.1
Move the limit inside the trig function because sine is continuous.
Step 6.1.3.1.2
Move the term outside of the limit because it is constant with respect to .
Step 6.1.3.2
Evaluate the limit of by plugging in for .
Step 6.1.3.3
Simplify the answer.
Step 6.1.3.3.1
Multiply by .
Step 6.1.3.3.2
The exact value of is .
Step 6.1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 6.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 6.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 6.2
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 6.3
Find the derivative of the numerator and denominator.
Step 6.3.1
Differentiate the numerator and denominator.
Step 6.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.3
Differentiate using the Power Rule which states that is where .
Step 6.3.4
Multiply by .
Step 6.3.5
Differentiate using the chain rule, which states that is where and .
Step 6.3.5.1
To apply the Chain Rule, set as .
Step 6.3.5.2
The derivative of with respect to is .
Step 6.3.5.3
Replace all occurrences of with .
Step 6.3.6
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.7
Differentiate using the Power Rule which states that is where .
Step 6.3.8
Multiply by .
Step 6.3.9
Move to the left of .
Step 6.4
Cancel the common factor of .
Step 6.4.1
Cancel the common factor.
Step 6.4.2
Rewrite the expression.
Step 6.5
Convert from to .
Step 6.6
Evaluate the limit.
Step 6.6.1
Move the limit inside the trig function because secant is continuous.
Step 6.6.2
Move the term outside of the limit because it is constant with respect to .
Step 6.7
Evaluate the limit of by plugging in for .
Step 6.8
Simplify the answer.
Step 6.8.1
Multiply by .
Step 6.8.2
The exact value of is .
Step 7
Step 7.1
Cancel the common factor.
Step 7.2
Rewrite the expression.
Step 8
Evaluate the limit of which is constant as approaches .
Step 9
Step 9.1
Multiply by .
Step 9.2
Multiply by .
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form: