Calculus Examples

Step 1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.2
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Step 1.3
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Step 1.4
Infinity divided by infinity 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
Find the derivative of the numerator and denominator.
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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
Evaluate .
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Multiply by .
Step 3.4
Evaluate .
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Step 3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Multiply by .
Step 3.5
By the Sum Rule, the derivative of with respect to is .
Step 3.6
Evaluate .
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Step 3.6.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.2
Differentiate using the Power Rule which states that is where .
Step 3.6.3
Multiply by .
Step 3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.8
Add and .
Step 4
Move the term outside of the limit because it is constant with respect to .
Step 5
Divide the numerator and denominator by the highest power of in the denominator, which is .
Step 6
Evaluate the limit.
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Step 6.1
Simplify each term.
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Step 6.1.1
Cancel the common factor of .
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Step 6.1.1.1
Cancel the common factor.
Step 6.1.1.2
Divide by .
Step 6.1.2
Cancel the common factor of and .
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Step 6.1.2.1
Factor out of .
Step 6.1.2.2
Cancel the common factors.
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Step 6.1.2.2.1
Factor out of .
Step 6.1.2.2.2
Cancel the common factor.
Step 6.1.2.2.3
Rewrite the expression.
Step 6.1.3
Move the negative in front of the fraction.
Step 6.2
Cancel the common factor of .
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Step 6.2.1
Cancel the common factor.
Step 6.2.2
Rewrite the expression.
Step 6.3
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 6.4
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 6.5
Evaluate the limit of which is constant as approaches .
Step 6.6
Move the term outside of the limit because it is constant with respect to .
Step 7
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Step 8
Evaluate the limit.
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Step 8.1
Evaluate the limit of which is constant as approaches .
Step 8.2
Simplify the answer.
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Step 8.2.1
Divide by .
Step 8.2.2
Multiply by .
Step 8.2.3
Add and .
Step 8.2.4
Cancel the common factor of .
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Step 8.2.4.1
Factor out of .
Step 8.2.4.2
Factor out of .
Step 8.2.4.3
Cancel the common factor.
Step 8.2.4.4
Rewrite the expression.
Step 8.2.5
Combine and .
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