Calculus Examples

Evaluate the Limit limit as x approaches 0 of (cos(x)-1+1/2x^2)/(x^4)
Step 1
Simplify terms.
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Step 1.1
Simplify the limit argument.
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Step 1.1.1
Combine and .
Step 1.1.2
Combine terms.
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Step 1.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.2
Combine and .
Step 1.1.2.3
Combine the numerators over the common denominator.
Step 1.2
Multiply by .
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
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.2
Move the limit inside the trig function because cosine is continuous.
Step 2.1.2.3
Move the term outside of the limit because it is constant with respect to .
Step 2.1.2.4
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.1.2.5
Evaluate the limit of which is constant as approaches .
Step 2.1.2.6
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.2.7
Evaluate the limits by plugging in for all occurrences of .
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Step 2.1.2.7.1
Evaluate the limit of by plugging in for .
Step 2.1.2.7.2
Evaluate the limit of by plugging in for .
Step 2.1.2.8
Simplify the answer.
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Step 2.1.2.8.1
Simplify each term.
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Step 2.1.2.8.1.1
The exact value of is .
Step 2.1.2.8.1.2
Raising to any positive power yields .
Step 2.1.2.8.1.3
Add and .
Step 2.1.2.8.1.4
Cancel the common factor of .
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Step 2.1.2.8.1.4.1
Factor out of .
Step 2.1.2.8.1.4.2
Cancel the common factor.
Step 2.1.2.8.1.4.3
Rewrite the expression.
Step 2.1.2.8.2
Subtract from .
Step 2.1.3
Evaluate the limit of the denominator.
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Step 2.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.1.3.2
Evaluate the limit of by plugging in for .
Step 2.1.3.3
Raising to any positive power yields .
Step 2.1.3.4
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
Rewrite as .
Step 2.3.3
Factor out of .
Step 2.3.4
Factor out of .
Step 2.3.5
Move the negative in front of the fraction.
Step 2.3.6
By the Sum Rule, the derivative of with respect to is .
Step 2.3.7
The derivative of with respect to is .
Step 2.3.8
Evaluate .
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Step 2.3.8.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3.8.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.8.5
Differentiate using the Power Rule which states that is where .
Step 2.3.8.6
Multiply by .
Step 2.3.8.7
Subtract from .
Step 2.3.8.8
Multiply by .
Step 2.3.8.9
Combine and .
Step 2.3.8.10
Combine and .
Step 2.3.8.11
Cancel the common factor of .
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Step 2.3.8.11.1
Cancel the common factor.
Step 2.3.8.11.2
Divide by .
Step 2.3.9
Reorder terms.
Step 2.3.10
Differentiate using the Power Rule which states that is where .
Step 3
Move the term outside of the limit because it is constant with respect to .
Step 4
Apply L'Hospital's rule.
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Step 4.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 4.1.1
Take the limit of the numerator and the limit of the denominator.
Step 4.1.2
Evaluate the limit of the numerator.
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Step 4.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 4.1.2.2
Move the limit inside the trig function because sine is continuous.
Step 4.1.2.3
Evaluate the limits by plugging in for all occurrences of .
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Step 4.1.2.3.1
Evaluate the limit of by plugging in for .
Step 4.1.2.3.2
Evaluate the limit of by plugging in for .
Step 4.1.2.4
Simplify the answer.
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Step 4.1.2.4.1
Simplify each term.
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Step 4.1.2.4.1.1
The exact value of is .
Step 4.1.2.4.1.2
Multiply by .
Step 4.1.2.4.2
Add and .
Step 4.1.3
Evaluate the limit of the denominator.
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Step 4.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 4.1.3.2
Evaluate the limit of by plugging in for .
Step 4.1.3.3
Raising to any positive power yields .
Step 4.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 4.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 4.2
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 4.3
Find the derivative of the numerator and denominator.
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Step 4.3.1
Differentiate the numerator and denominator.
Step 4.3.2
By the Sum Rule, the derivative of with respect to is .
Step 4.3.3
Differentiate using the Power Rule which states that is where .
Step 4.3.4
Evaluate .
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Step 4.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.4.2
The derivative of with respect to is .
Step 4.3.5
Differentiate using the Power Rule which states that is where .
Step 5
Move the term outside of the limit because it is constant with respect to .
Step 6
Apply L'Hospital's rule.
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Step 6.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 6.1.1
Take the limit of the numerator and the limit of the denominator.
Step 6.1.2
Evaluate the limit of the numerator.
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Step 6.1.2.1
Evaluate the limit.
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Step 6.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 6.1.2.1.2
Evaluate the limit of which is constant as approaches .
Step 6.1.2.1.3
Move the limit inside the trig function because cosine is continuous.
Step 6.1.2.2
Evaluate the limit of by plugging in for .
Step 6.1.2.3
Simplify the answer.
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Step 6.1.2.3.1
Simplify each term.
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Step 6.1.2.3.1.1
The exact value of is .
Step 6.1.2.3.1.2
Multiply by .
Step 6.1.2.3.2
Subtract from .
Step 6.1.3
Evaluate the limit of the denominator.
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Step 6.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 6.1.3.2
Evaluate the limit of by plugging in for .
Step 6.1.3.3
Raising to any positive power yields .
Step 6.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 6.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 6.2
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 6.3
Find the derivative of the numerator and denominator.
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Step 6.3.1
Differentiate the numerator and denominator.
Step 6.3.2
By the Sum Rule, the derivative of with respect to is .
Step 6.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.4
Evaluate .
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Step 6.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.4.2
The derivative of with respect to is .
Step 6.3.4.3
Multiply by .
Step 6.3.4.4
Multiply by .
Step 6.3.5
Add and .
Step 6.3.6
Differentiate using the Power Rule which states that is where .
Step 7
Move the term outside of the limit because it is constant with respect to .
Step 8
Since and , apply the squeeze theorem.
Step 9
Simplify the answer.
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Step 9.1
Multiply .
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Step 9.1.1
Multiply by .
Step 9.1.2
Multiply by .
Step 9.2
Multiply .
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Step 9.2.1
Multiply by .
Step 9.2.2
Multiply by .
Step 9.3
Multiply by .
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form: