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Calculus Examples
Step 1
Move the term outside of the limit because it is constant with respect to .
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 by plugging in for .
Step 2.1.3
Evaluate the limit of the denominator.
Step 2.1.3.1
Evaluate the limit.
Step 2.1.3.1.1
Move the limit inside the trig function because tangent is continuous.
Step 2.1.3.1.2
Move the term outside of the limit because it is constant with respect to .
Step 2.1.3.2
Evaluate the limit of by plugging in for .
Step 2.1.3.3
Simplify the answer.
Step 2.1.3.3.1
Multiply by .
Step 2.1.3.3.2
The exact value of is .
Step 2.1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
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
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Differentiate using the chain rule, which states that is where and .
Step 2.3.3.1
To apply the Chain Rule, set as .
Step 2.3.3.2
The derivative of with respect to is .
Step 2.3.3.3
Replace all occurrences of with .
Step 2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.5
Differentiate using the Power Rule which states that is where .
Step 2.3.6
Multiply by .
Step 2.3.7
Move to the left of .
Step 3
Step 3.1
Move the term outside of the limit because it is constant with respect to .
Step 3.2
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 3.3
Evaluate the limit of which is constant as approaches .
Step 3.4
Move the exponent from outside the limit using the Limits Power Rule.
Step 3.5
Move the limit inside the trig function because secant is continuous.
Step 3.6
Move the term outside of the limit because it is constant with respect to .
Step 4
Evaluate the limit of by plugging in for .
Step 5
Step 5.1
Combine and .
Step 5.2
Combine.
Step 5.3
Multiply by .
Step 5.4
Simplify the denominator.
Step 5.4.1
Multiply by .
Step 5.4.2
The exact value of is .
Step 5.4.3
One to any power is one.
Step 5.5
Multiply by .
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form: