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Calculus Examples
Step 1
Step 1.1
Simplify the limit argument.
Step 1.1.1
Combine and .
Step 1.1.2
Combine terms.
Step 1.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.2
Combine and .
Step 1.1.2.3
Combine the numerators over the common denominator.
Step 1.2
Multiply by .
Step 2
Step 2.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 2.1.1
Take the limit of the numerator and the limit of the denominator.
Step 2.1.2
Evaluate the limit of the numerator.
Step 2.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.2
Move the limit inside the trig function because cosine is continuous.
Step 2.1.2.3
Move the term outside of the limit because it is constant with respect to .
Step 2.1.2.4
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.5
Evaluate the limit of which is constant as approaches .
Step 2.1.2.6
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.2.7
Evaluate the limits by plugging in for all occurrences of .
Step 2.1.2.7.1
Evaluate the limit of by plugging in for .
Step 2.1.2.7.2
Evaluate the limit of by plugging in for .
Step 2.1.2.8
Simplify the answer.
Step 2.1.2.8.1
Simplify each term.
Step 2.1.2.8.1.1
The exact value of is .
Step 2.1.2.8.1.2
Raising to any positive power yields .
Step 2.1.2.8.1.3
Add and .
Step 2.1.2.8.1.4
Cancel the common factor of .
Step 2.1.2.8.1.4.1
Factor out of .
Step 2.1.2.8.1.4.2
Cancel the common factor.
Step 2.1.2.8.1.4.3
Rewrite the expression.
Step 2.1.2.8.2
Subtract from .
Step 2.1.3
Evaluate the limit of the denominator.
Step 2.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.3.2
Evaluate the limit of by plugging in for .
Step 2.1.3.3
Raising to any positive power yields .
Step 2.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 2.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 2.2
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 2.3
Find the derivative of the numerator and denominator.
Step 2.3.1
Differentiate the numerator and denominator.
Step 2.3.2
Rewrite as .
Step 2.3.3
Factor out of .
Step 2.3.4
Factor out of .
Step 2.3.5
Move the negative in front of the fraction.
Step 2.3.6
By the Sum Rule, the derivative of with respect to is .
Step 2.3.7
The derivative of with respect to is .
Step 2.3.8
Evaluate .
Step 2.3.8.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3.8.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.5
Differentiate using the Power Rule which states that is where .
Step 2.3.8.6
Multiply by .
Step 2.3.8.7
Subtract from .
Step 2.3.8.8
Multiply by .
Step 2.3.8.9
Combine and .
Step 2.3.8.10
Combine and .
Step 2.3.8.11
Cancel the common factor of .
Step 2.3.8.11.1
Cancel the common factor.
Step 2.3.8.11.2
Divide by .
Step 2.3.9
Reorder terms.
Step 2.3.10
Differentiate using the Power Rule which states that is where .
Step 3
Move the term outside of the limit because it is constant with respect to .
Step 4
Step 4.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 4.1.1
Take the limit of the numerator and the limit of the denominator.
Step 4.1.2
Evaluate the limit of the numerator.
Step 4.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 4.1.2.2
Move the limit inside the trig function because sine is continuous.
Step 4.1.2.3
Evaluate the limits by plugging in for all occurrences of .
Step 4.1.2.3.1
Evaluate the limit of by plugging in for .
Step 4.1.2.3.2
Evaluate the limit of by plugging in for .
Step 4.1.2.4
Simplify the answer.
Step 4.1.2.4.1
Simplify each term.
Step 4.1.2.4.1.1
The exact value of is .
Step 4.1.2.4.1.2
Multiply by .
Step 4.1.2.4.2
Add and .
Step 4.1.3
Evaluate the limit of the denominator.
Step 4.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 4.1.3.2
Evaluate the limit of by plugging in for .
Step 4.1.3.3
Raising to any positive power yields .
Step 4.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 4.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 4.2
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 4.3
Find the derivative of the numerator and denominator.
Step 4.3.1
Differentiate the numerator and denominator.
Step 4.3.2
By the Sum Rule, the derivative of with respect to is .
Step 4.3.3
Differentiate using the Power Rule which states that is where .
Step 4.3.4
Evaluate .
Step 4.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.4.2
The derivative of with respect to is .
Step 4.3.5
Differentiate using the Power Rule which states that is where .
Step 5
Move the term outside of the limit because it is constant with respect to .
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
Evaluate the limit.
Step 6.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 6.1.2.1.2
Evaluate the limit of which is constant as approaches .
Step 6.1.2.1.3
Move the limit inside the trig function because cosine is continuous.
Step 6.1.2.2
Evaluate the limit of by plugging in for .
Step 6.1.2.3
Simplify the answer.
Step 6.1.2.3.1
Simplify each term.
Step 6.1.2.3.1.1
The exact value of is .
Step 6.1.2.3.1.2
Multiply by .
Step 6.1.2.3.2
Subtract from .
Step 6.1.3
Evaluate the limit of the denominator.
Step 6.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 6.1.3.2
Evaluate the limit of by plugging in for .
Step 6.1.3.3
Raising to any positive power yields .
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
By the Sum Rule, the derivative of with respect to is .
Step 6.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.4
Evaluate .
Step 6.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.4.2
The derivative of with respect to is .
Step 6.3.4.3
Multiply by .
Step 6.3.4.4
Multiply by .
Step 6.3.5
Add and .
Step 6.3.6
Differentiate using the Power Rule which states that is where .
Step 7
Move the term outside of the limit because it is constant with respect to .
Step 8
Since and , apply the squeeze theorem.
Step 9
Step 9.1
Multiply .
Step 9.1.1
Multiply by .
Step 9.1.2
Multiply by .
Step 9.2
Multiply .
Step 9.2.1
Multiply by .
Step 9.2.2
Multiply by .
Step 9.3
Multiply by .
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form: