Calculus Examples

Evaluate the Limit limit as x approaches 0 of ( square root of 2+x- square root of 2)/x
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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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.
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Step 1.1.2.1
Evaluate the limit.
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Step 1.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.1.2
Move the limit under the radical sign.
Step 1.1.2.1.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.1.4
Evaluate the limit of which is constant as approaches .
Step 1.1.2.1.5
Evaluate the limit of which is constant as approaches .
Step 1.1.2.2
Evaluate the limit of by plugging in for .
Step 1.1.2.3
Simplify the answer.
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Step 1.1.2.3.1
Add and .
Step 1.1.2.3.2
Subtract from .
Step 1.1.3
Evaluate the limit of by plugging in for .
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.
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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 .
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Step 1.3.3.1
Use to rewrite as .
Step 1.3.3.2
Differentiate using the chain rule, which states that is where and .
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Step 1.3.3.2.1
To apply the Chain Rule, set as .
Step 1.3.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3.2.3
Replace all occurrences of with .
Step 1.3.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3.5
Differentiate using the Power Rule which states that is where .
Step 1.3.3.6
To write as a fraction with a common denominator, multiply by .
Step 1.3.3.7
Combine and .
Step 1.3.3.8
Combine the numerators over the common denominator.
Step 1.3.3.9
Simplify the numerator.
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Step 1.3.3.9.1
Multiply by .
Step 1.3.3.9.2
Subtract from .
Step 1.3.3.10
Move the negative in front of the fraction.
Step 1.3.3.11
Add and .
Step 1.3.3.12
Combine and .
Step 1.3.3.13
Multiply by .
Step 1.3.3.14
Move to the denominator using the negative exponent rule .
Step 1.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.5
Add and .
Step 1.3.6
Differentiate using the Power Rule which states that is where .
Step 1.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.5
Rewrite as .
Step 1.6
Multiply by .
Step 2
Evaluate the limit.
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Step 2.1
Move the term outside of the limit because it is constant with respect to .
Step 2.2
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.3
Evaluate the limit of which is constant as approaches .
Step 2.4
Move the limit under the radical sign.
Step 2.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.6
Evaluate the limit of which is constant as approaches .
Step 3
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Add and .
Step 4.2
Multiply by .
Step 4.3
Combine and simplify the denominator.
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Step 4.3.1
Multiply by .
Step 4.3.2
Raise to the power of .
Step 4.3.3
Raise to the power of .
Step 4.3.4
Use the power rule to combine exponents.
Step 4.3.5
Add and .
Step 4.3.6
Rewrite as .
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Step 4.3.6.1
Use to rewrite as .
Step 4.3.6.2
Apply the power rule and multiply exponents, .
Step 4.3.6.3
Combine and .
Step 4.3.6.4
Cancel the common factor of .
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Step 4.3.6.4.1
Cancel the common factor.
Step 4.3.6.4.2
Rewrite the expression.
Step 4.3.6.5
Evaluate the exponent.
Step 4.4
Multiply .
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Step 4.4.1
Multiply by .
Step 4.4.2
Multiply by .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form: