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Calculus Examples
Step 1
Step 1.1
To write as a fraction with a common denominator, multiply by .
Step 1.2
Multiply by .
Step 1.3
Combine the numerators over the common denominator.
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
Evaluate the limit.
Step 2.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.1.2
Evaluate the limit of which is constant as approaches .
Step 2.1.2.1.3
Move the term outside of the limit because it is constant with respect to .
Step 2.1.2.1.4
Move the limit under the radical sign.
Step 2.1.2.1.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.1.6
Evaluate the limit of which is constant as approaches .
Step 2.1.2.2
Evaluate the limit of by plugging in for .
Step 2.1.2.3
Simplify the answer.
Step 2.1.2.3.1
Simplify each term.
Step 2.1.2.3.1.1
Add and .
Step 2.1.2.3.1.2
Any root of is .
Step 2.1.2.3.1.3
Multiply by .
Step 2.1.2.3.2
Subtract from .
Step 2.1.3
Evaluate the limit of the denominator.
Step 2.1.3.1
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 2.1.3.2
Move the limit under the radical sign.
Step 2.1.3.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.3.4
Evaluate the limit of which is constant as approaches .
Step 2.1.3.5
Evaluate the limits by plugging in for all occurrences of .
Step 2.1.3.5.1
Evaluate the limit of by plugging in for .
Step 2.1.3.5.2
Evaluate the limit of by plugging in for .
Step 2.1.3.6
Simplify the answer.
Step 2.1.3.6.1
Add and .
Step 2.1.3.6.2
Any root of is .
Step 2.1.3.6.3
Multiply by .
Step 2.1.3.6.4
The expression contains a division by . The expression is undefined.
Undefined
Step 2.1.3.7
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
By the Sum Rule, the derivative of with respect to is .
Step 2.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.4
Evaluate .
Step 2.3.4.1
Use to rewrite as .
Step 2.3.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.4.3
Differentiate using the chain rule, which states that is where and .
Step 2.3.4.3.1
To apply the Chain Rule, set as .
Step 2.3.4.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.4.3.3
Replace all occurrences of with .
Step 2.3.4.4
By the Sum Rule, the derivative of with respect to is .
Step 2.3.4.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.4.6
Differentiate using the Power Rule which states that is where .
Step 2.3.4.7
To write as a fraction with a common denominator, multiply by .
Step 2.3.4.8
Combine and .
Step 2.3.4.9
Combine the numerators over the common denominator.
Step 2.3.4.10
Simplify the numerator.
Step 2.3.4.10.1
Multiply by .
Step 2.3.4.10.2
Subtract from .
Step 2.3.4.11
Move the negative in front of the fraction.
Step 2.3.4.12
Add and .
Step 2.3.4.13
Combine and .
Step 2.3.4.14
Multiply by .
Step 2.3.4.15
Move to the denominator using the negative exponent rule .
Step 2.3.4.16
Combine and .
Step 2.3.4.17
Move the negative in front of the fraction.
Step 2.3.5
Subtract from .
Step 2.3.6
Use to rewrite as .
Step 2.3.7
Differentiate using the Product Rule which states that is where and .
Step 2.3.8
Differentiate using the chain rule, which states that is where and .
Step 2.3.8.1
To apply the Chain Rule, set as .
Step 2.3.8.2
Differentiate using the Power Rule which states that is where .
Step 2.3.8.3
Replace all occurrences of with .
Step 2.3.9
To write as a fraction with a common denominator, multiply by .
Step 2.3.10
Combine and .
Step 2.3.11
Combine the numerators over the common denominator.
Step 2.3.12
Simplify the numerator.
Step 2.3.12.1
Multiply by .
Step 2.3.12.2
Subtract from .
Step 2.3.13
Move the negative in front of the fraction.
Step 2.3.14
Combine and .
Step 2.3.15
Move to the denominator using the negative exponent rule .
Step 2.3.16
Combine and .
Step 2.3.17
By the Sum Rule, the derivative of with respect to is .
Step 2.3.18
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.19
Add and .
Step 2.3.20
Differentiate using the Power Rule which states that is where .
Step 2.3.21
Multiply by .
Step 2.3.22
Differentiate using the Power Rule which states that is where .
Step 2.3.23
Multiply by .
Step 2.3.24
To write as a fraction with a common denominator, multiply by .
Step 2.3.25
Combine and .
Step 2.3.26
Combine the numerators over the common denominator.
Step 2.3.27
Multiply by by adding the exponents.
Step 2.3.27.1
Move .
Step 2.3.27.2
Use the power rule to combine exponents.
Step 2.3.27.3
Combine the numerators over the common denominator.
Step 2.3.27.4
Add and .
Step 2.3.27.5
Divide by .
Step 2.3.28
Simplify .
Step 2.3.29
Move to the left of .
Step 2.3.30
Simplify.
Step 2.3.30.1
Apply the distributive property.
Step 2.3.30.2
Simplify the numerator.
Step 2.3.30.2.1
Multiply by .
Step 2.3.30.2.2
Add and .
Step 2.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.5
Convert fractional exponents to radicals.
Step 2.5.1
Rewrite as .
Step 2.5.2
Rewrite as .
Step 2.6
Combine factors.
Step 2.6.1
Multiply by .
Step 2.6.2
Multiply by .
Step 2.7
Reduce.
Step 2.7.1
Cancel the common factor of and .
Step 2.7.1.1
Factor out of .
Step 2.7.1.2
Cancel the common factors.
Step 2.7.1.2.1
Factor out of .
Step 2.7.1.2.2
Cancel the common factor.
Step 2.7.1.2.3
Rewrite the expression.
Step 2.7.2
Cancel the common factor of .
Step 2.7.2.1
Cancel the common factor.
Step 2.7.2.2
Rewrite the expression.
Step 3
Step 3.1
Move the term outside of the limit because it is constant with respect to .
Step 3.2
Move the term outside of the limit because it is constant with respect to .
Step 3.3
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 3.4
Evaluate the limit of which is constant as approaches .
Step 3.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3.6
Move the term outside of the limit because it is constant with respect to .
Step 3.7
Evaluate the limit of which is constant as approaches .
Step 4
Evaluate the limit of by plugging in for .
Step 5
Step 5.1
Simplify the denominator.
Step 5.1.1
Multiply by .
Step 5.1.2
Add and .
Step 5.2
Combine and .
Step 5.3
Move the negative in front of the fraction.
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form: