Calculus Examples

Evaluate the Limit limit as t approaches 0 of 5/(t square root of 1+t)-5/t
Step 1
Combine terms.
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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
Apply L'Hospital's rule.
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Step 2.1
Evaluate the limit of the numerator and the limit of the denominator.
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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.
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Step 2.1.2.1
Evaluate the limit.
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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.
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Step 2.1.2.3.1
Simplify each term.
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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.
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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 .
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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.
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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.
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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 .
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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 .
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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.
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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 .
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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.
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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.
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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.
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Step 2.3.30.1
Apply the distributive property.
Step 2.3.30.2
Simplify the numerator.
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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.
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Step 2.5.1
Rewrite as .
Step 2.5.2
Rewrite as .
Step 2.6
Combine factors.
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Step 2.6.1
Multiply by .
Step 2.6.2
Multiply by .
Step 2.7
Reduce.
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Step 2.7.1
Cancel the common factor of and .
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Step 2.7.1.1
Factor out of .
Step 2.7.1.2
Cancel the common factors.
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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 .
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Step 2.7.2.1
Cancel the common factor.
Step 2.7.2.2
Rewrite the expression.
Step 3
Evaluate the limit.
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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
Simplify the answer.
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Step 5.1
Simplify the denominator.
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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: