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Calculus Examples
Step 1
Step 1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.2
Evaluate the limit of the numerator.
Step 1.2.1
Evaluate the limit.
Step 1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.2.1.2
Move the limit into the exponent.
Step 1.2.1.3
Evaluate the limit of which is constant as approaches .
Step 1.2.2
Evaluate the limit of by plugging in for .
Step 1.2.3
Simplify the answer.
Step 1.2.3.1
Simplify each term.
Step 1.2.3.1.1
Anything raised to is .
Step 1.2.3.1.2
Multiply by .
Step 1.2.3.2
Subtract from .
Step 1.3
Evaluate the limit of the denominator.
Step 1.3.1
Evaluate the limit.
Step 1.3.1.1
Move the limit inside the trig function because sine is continuous.
Step 1.3.1.2
Move the term outside of the limit because it is constant with respect to .
Step 1.3.2
Evaluate the limit of by plugging in for .
Step 1.3.3
Simplify the answer.
Step 1.3.3.1
Multiply by .
Step 1.3.3.2
The exact value of is .
Step 1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 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 3
Step 3.1
Differentiate the numerator and denominator.
Step 3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3
Differentiate using the Exponential Rule which states that is where =.
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Add and .
Step 3.6
Differentiate using the chain rule, which states that is where and .
Step 3.6.1
To apply the Chain Rule, set as .
Step 3.6.2
The derivative of with respect to is .
Step 3.6.3
Replace all occurrences of with .
Step 3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.8
Differentiate using the Power Rule which states that is where .
Step 3.9
Multiply by .
Step 3.10
Move to the left of .
Step 3.11
Multiply by .
Step 4
Move the term outside of the limit because it is constant with respect to .
Step 5
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 6
Move the limit into the exponent.
Step 7
Move the limit inside the trig function because cosine is continuous.
Step 8
Move the term outside of the limit because it is constant with respect to .
Step 9
Step 9.1
Evaluate the limit of by plugging in for .
Step 9.2
Evaluate the limit of by plugging in for .
Step 10
Step 10.1
Combine.
Step 10.2
Multiply by .
Step 10.3
Simplify the denominator.
Step 10.3.1
Multiply by .
Step 10.3.2
The exact value of is .
Step 10.4
Anything raised to is .
Step 10.5
Multiply by .