Calculus Examples

Evaluate the Limit limit as x approaches 10 of (x-10)/( square root of x- square root of 20-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
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
Multiply by .
Step 1.1.2.3.2
Subtract from .
Step 1.1.3
Evaluate the limit of the denominator.
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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 limit under the radical sign.
Step 1.1.3.3
Move the limit under the radical sign.
Step 1.1.3.4
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.3.5
Evaluate the limit of which is constant as approaches .
Step 1.1.3.6
Evaluate the limits by plugging in for all occurrences of .
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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.
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Step 1.1.3.7.1
Subtract from .
Step 1.1.3.7.2
Subtract from .
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.
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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
Differentiate using the Power Rule which states that is where .
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
By the Sum Rule, the derivative of with respect to is .
Step 1.3.7
Evaluate .
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Step 1.3.7.1
Use to rewrite as .
Step 1.3.7.2
Differentiate using the Power Rule which states that is where .
Step 1.3.7.3
To write as a fraction with a common denominator, multiply by .
Step 1.3.7.4
Combine and .
Step 1.3.7.5
Combine the numerators over the common denominator.
Step 1.3.7.6
Simplify the numerator.
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Step 1.3.7.6.1
Multiply by .
Step 1.3.7.6.2
Subtract from .
Step 1.3.7.7
Move the negative in front of the fraction.
Step 1.3.8
Evaluate .
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Step 1.3.8.1
Use to rewrite as .
Step 1.3.8.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.8.3
Differentiate using the chain rule, which states that is where and .
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Step 1.3.8.3.1
To apply the Chain Rule, set as .
Step 1.3.8.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.8.3.3
Replace all occurrences of with .
Step 1.3.8.4
By the Sum Rule, the derivative of with respect to is .
Step 1.3.8.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.8.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.8.7
Differentiate using the Power Rule which states that is where .
Step 1.3.8.8
To write as a fraction with a common denominator, multiply by .
Step 1.3.8.9
Combine and .
Step 1.3.8.10
Combine the numerators over the common denominator.
Step 1.3.8.11
Simplify the numerator.
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Step 1.3.8.11.1
Multiply by .
Step 1.3.8.11.2
Subtract from .
Step 1.3.8.12
Move the negative in front of the fraction.
Step 1.3.8.13
Multiply by .
Step 1.3.8.14
Subtract from .
Step 1.3.8.15
Combine and .
Step 1.3.8.16
Combine and .
Step 1.3.8.17
Move to the left of .
Step 1.3.8.18
Rewrite as .
Step 1.3.8.19
Move to the denominator using the negative exponent rule .
Step 1.3.8.20
Move the negative in front of the fraction.
Step 1.3.8.21
Multiply by .
Step 1.3.8.22
Multiply by .
Step 1.3.9
Simplify.
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Step 1.3.9.1
Rewrite the expression using the negative exponent rule .
Step 1.3.9.2
Multiply by .
Step 1.4
Convert fractional exponents to radicals.
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Step 1.4.1
Rewrite as .
Step 1.4.2
Rewrite as .
Step 2
Evaluate the limit.
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Step 2.1
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.2
Evaluate the limit of which is constant as approaches .
Step 2.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.4
Move the term outside of the limit because it is constant with respect to .
Step 2.5
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.6
Evaluate the limit of which is constant as approaches .
Step 2.7
Move the limit under the radical sign.
Step 2.8
Move the term outside of the limit because it is constant with respect to .
Step 2.9
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.10
Evaluate the limit of which is constant as approaches .
Step 2.11
Move the limit under the radical sign.
Step 2.12
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.13
Evaluate the limit of which is constant as approaches .
Step 3
Evaluate the limits by plugging in for all occurrences of .
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Step 3.1
Evaluate the limit of by plugging in for .
Step 3.2
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Subtract from .
Step 4.2
Simplify the denominator.
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Step 4.2.1
Multiply by .
Step 4.2.2
Combine and simplify the denominator.
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Step 4.2.2.1
Multiply by .
Step 4.2.2.2
Raise to the power of .
Step 4.2.2.3
Raise to the power of .
Step 4.2.2.4
Use the power rule to combine exponents.
Step 4.2.2.5
Add and .
Step 4.2.2.6
Rewrite as .
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Step 4.2.2.6.1
Use to rewrite as .
Step 4.2.2.6.2
Apply the power rule and multiply exponents, .
Step 4.2.2.6.3
Combine and .
Step 4.2.2.6.4
Cancel the common factor of .
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Step 4.2.2.6.4.1
Cancel the common factor.
Step 4.2.2.6.4.2
Rewrite the expression.
Step 4.2.2.6.5
Evaluate the exponent.
Step 4.2.3
Multiply .
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Step 4.2.3.1
Multiply by .
Step 4.2.3.2
Multiply by .
Step 4.2.4
Multiply by .
Step 4.2.5
Combine and simplify the denominator.
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Step 4.2.5.1
Multiply by .
Step 4.2.5.2
Raise to the power of .
Step 4.2.5.3
Raise to the power of .
Step 4.2.5.4
Use the power rule to combine exponents.
Step 4.2.5.5
Add and .
Step 4.2.5.6
Rewrite as .
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Step 4.2.5.6.1
Use to rewrite as .
Step 4.2.5.6.2
Apply the power rule and multiply exponents, .
Step 4.2.5.6.3
Combine and .
Step 4.2.5.6.4
Cancel the common factor of .
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Step 4.2.5.6.4.1
Cancel the common factor.
Step 4.2.5.6.4.2
Rewrite the expression.
Step 4.2.5.6.5
Evaluate the exponent.
Step 4.2.6
Multiply .
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Step 4.2.6.1
Multiply by .
Step 4.2.6.2
Multiply by .
Step 4.2.7
Combine the numerators over the common denominator.
Step 4.2.8
Rewrite in a factored form.
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Step 4.2.8.1
Add and .
Step 4.2.8.2
Reduce the expression by cancelling the common factors.
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Step 4.2.8.2.1
Factor out of .
Step 4.2.8.2.2
Factor out of .
Step 4.2.8.2.3
Cancel the common factor.
Step 4.2.8.2.4
Rewrite the expression.
Step 4.3
Multiply the numerator by the reciprocal of the denominator.
Step 4.4
Multiply by .
Step 4.5
Multiply by .
Step 4.6
Combine and simplify the denominator.
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Step 4.6.1
Multiply by .
Step 4.6.2
Raise to the power of .
Step 4.6.3
Raise to the power of .
Step 4.6.4
Use the power rule to combine exponents.
Step 4.6.5
Add and .
Step 4.6.6
Rewrite as .
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Step 4.6.6.1
Use to rewrite as .
Step 4.6.6.2
Apply the power rule and multiply exponents, .
Step 4.6.6.3
Combine and .
Step 4.6.6.4
Cancel the common factor of .
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Step 4.6.6.4.1
Cancel the common factor.
Step 4.6.6.4.2
Rewrite the expression.
Step 4.6.6.5
Evaluate the exponent.
Step 4.7
Cancel the common factor of .
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Step 4.7.1
Cancel the common factor.
Step 4.7.2
Divide by .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form: