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Calculus Examples
Step 1
Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.1.2
Evaluate the limit of the numerator.
Step 1.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.2
Move the term outside of the limit because it is constant with respect to .
Step 1.1.2.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.2.4
Move the term outside of the limit because it is constant with respect to .
Step 1.1.2.5
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.2.6
Evaluate the limits by plugging in for all occurrences of .
Step 1.1.2.6.1
Evaluate the limit of by plugging in for .
Step 1.1.2.6.2
Evaluate the limit of by plugging in for .
Step 1.1.2.7
Simplify the answer.
Step 1.1.2.7.1
Simplify each term.
Step 1.1.2.7.1.1
Raising to any positive power yields .
Step 1.1.2.7.1.2
Multiply by .
Step 1.1.2.7.1.3
Raising to any positive power yields .
Step 1.1.2.7.1.4
Multiply by .
Step 1.1.2.7.2
Add and .
Step 1.1.3
Evaluate the limit of the denominator.
Step 1.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.3.2
Move the term outside of the limit because it is constant with respect to .
Step 1.1.3.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.3.4
Move the term outside of the limit because it is constant with respect to .
Step 1.1.3.5
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.3.6
Evaluate the limits by plugging in for all occurrences of .
Step 1.1.3.6.1
Evaluate the limit of by plugging in for .
Step 1.1.3.6.2
Evaluate the limit of by plugging in for .
Step 1.1.3.7
Simplify the answer.
Step 1.1.3.7.1
Simplify each term.
Step 1.1.3.7.1.1
Raising to any positive power yields .
Step 1.1.3.7.1.2
Multiply by .
Step 1.1.3.7.1.3
Raising to any positive power yields .
Step 1.1.3.7.1.4
Multiply by .
Step 1.1.3.7.2
Add and .
Step 1.1.3.7.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.3.8
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.2
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 1.3
Find the derivative of the numerator and denominator.
Step 1.3.1
Differentiate the numerator and denominator.
Step 1.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.3
Evaluate .
Step 1.3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3.3
Multiply by .
Step 1.3.4
Evaluate .
Step 1.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4.2
Differentiate using the Power Rule which states that is where .
Step 1.3.4.3
Multiply by .
Step 1.3.5
By the Sum Rule, the derivative of with respect to is .
Step 1.3.6
Evaluate .
Step 1.3.6.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.6.2
Differentiate using the Power Rule which states that is where .
Step 1.3.6.3
Multiply by .
Step 1.3.7
Evaluate .
Step 1.3.7.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.7.2
Differentiate using the Power Rule which states that is where .
Step 1.3.7.3
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 term outside of the limit because it is constant with respect to .
Step 2.1.2.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.2.4
Move the term outside of the limit because it is constant with respect to .
Step 2.1.2.5
Evaluate the limits by plugging in for all occurrences of .
Step 2.1.2.5.1
Evaluate the limit of by plugging in for .
Step 2.1.2.5.2
Evaluate the limit of by plugging in for .
Step 2.1.2.6
Simplify the answer.
Step 2.1.2.6.1
Simplify each term.
Step 2.1.2.6.1.1
Raising to any positive power yields .
Step 2.1.2.6.1.2
Multiply by .
Step 2.1.2.6.1.3
Multiply by .
Step 2.1.2.6.2
Add and .
Step 2.1.3
Evaluate the limit of the denominator.
Step 2.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.3.2
Move the term outside of the limit because it is constant with respect to .
Step 2.1.3.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.3.4
Move the term outside of the limit because it is constant with respect to .
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
Simplify each term.
Step 2.1.3.6.1.1
Raising to any positive power yields .
Step 2.1.3.6.1.2
Multiply by .
Step 2.1.3.6.1.3
Multiply by .
Step 2.1.3.6.2
Add and .
Step 2.1.3.6.3
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
Evaluate .
Step 2.3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3.3
Multiply by .
Step 2.3.4
Evaluate .
Step 2.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.4.2
Differentiate using the Power Rule which states that is where .
Step 2.3.4.3
Multiply by .
Step 2.3.5
By the Sum Rule, the derivative of with respect to is .
Step 2.3.6
Evaluate .
Step 2.3.6.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.6.2
Differentiate using the Power Rule which states that is where .
Step 2.3.6.3
Multiply by .
Step 2.3.7
Evaluate .
Step 2.3.7.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.7.2
Differentiate using the Power Rule which states that is where .
Step 2.3.7.3
Multiply by .
Step 2.4
Cancel the common factor of and .
Step 2.4.1
Factor out of .
Step 2.4.2
Factor out of .
Step 2.4.3
Factor out of .
Step 2.4.4
Cancel the common factors.
Step 2.4.4.1
Factor out of .
Step 2.4.4.2
Factor out of .
Step 2.4.4.3
Factor out of .
Step 2.4.4.4
Cancel the common factor.
Step 2.4.4.5
Rewrite the expression.
Step 3
Step 3.1
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 3.2
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3.3
Move the term outside of the limit because it is constant with respect to .
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
Move the exponent from outside the limit using the Limits Power Rule.
Step 3.8
Evaluate the limit of which is constant as approaches .
Step 4
Step 4.1
Evaluate the limit of by plugging in for .
Step 4.2
Evaluate the limit of by plugging in for .
Step 5
Step 5.1
Cancel the common factor of and .
Step 5.1.1
Factor out of .
Step 5.1.2
Factor out of .
Step 5.1.3
Factor out of .
Step 5.1.4
Rewrite as .
Step 5.1.5
Factor out of .
Step 5.1.6
Factor out of .
Step 5.1.7
Factor out of .
Step 5.1.8
Cancel the common factors.
Step 5.1.8.1
Factor out of .
Step 5.1.8.2
Cancel the common factor.
Step 5.1.8.3
Rewrite the expression.
Step 5.2
Simplify the numerator.
Step 5.2.1
Multiply by .
Step 5.2.2
Add and .
Step 5.3
Simplify the denominator.
Step 5.3.1
Raising to any positive power yields .
Step 5.3.2
Multiply by .
Step 5.3.3
Add and .
Step 5.4
Multiply by .
Step 5.5
Cancel the common factor of and .
Step 5.5.1
Factor out of .
Step 5.5.2
Cancel the common factors.
Step 5.5.2.1
Factor out of .
Step 5.5.2.2
Cancel the common factor.
Step 5.5.2.3
Rewrite the expression.
Step 5.6
Move the negative in front of the fraction.
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form: