Calculus Examples

Evaluate the Limit limit as x approaches 0 of (sin(5x))/(sin(2x))
Step 1
Multiply the numerator and denominator by .
Step 2
Multiply the numerator and denominator by .
Step 3
Separate fractions.
Step 4
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 5
The limit of as approaches is .
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Step 5.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 5.1.1
Take the limit of the numerator and the limit of the denominator.
Step 5.1.2
Evaluate the limit of the numerator.
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Step 5.1.2.1
Evaluate the limit.
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Step 5.1.2.1.1
Move the limit inside the trig function because sine is continuous.
Step 5.1.2.1.2
Move the term outside of the limit because it is constant with respect to .
Step 5.1.2.2
Evaluate the limit of by plugging in for .
Step 5.1.2.3
Simplify the answer.
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Step 5.1.2.3.1
Multiply by .
Step 5.1.2.3.2
The exact value of is .
Step 5.1.3
Evaluate the limit of the denominator.
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Step 5.1.3.1
Move the term outside of the limit because it is constant with respect to .
Step 5.1.3.2
Evaluate the limit of by plugging in for .
Step 5.1.3.3
Multiply by .
Step 5.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.2
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 5.3
Find the derivative of the numerator and denominator.
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Step 5.3.1
Differentiate the numerator and denominator.
Step 5.3.2
Differentiate using the chain rule, which states that is where and .
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Step 5.3.2.1
To apply the Chain Rule, set as .
Step 5.3.2.2
The derivative of with respect to is .
Step 5.3.2.3
Replace all occurrences of with .
Step 5.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.4
Differentiate using the Power Rule which states that is where .
Step 5.3.5
Multiply by .
Step 5.3.6
Move to the left of .
Step 5.3.7
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.8
Differentiate using the Power Rule which states that is where .
Step 5.3.9
Multiply by .
Step 5.4
Evaluate the limit.
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Step 5.4.1
Cancel the common factor of .
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Step 5.4.1.1
Cancel the common factor.
Step 5.4.1.2
Divide by .
Step 5.4.2
Move the limit inside the trig function because cosine is continuous.
Step 5.4.3
Move the term outside of the limit because it is constant with respect to .
Step 5.5
Evaluate the limit of by plugging in for .
Step 5.6
Simplify the answer.
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Step 5.6.1
Multiply by .
Step 5.6.2
The exact value of is .
Step 6
The limit of as approaches is .
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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
Move the term outside of the limit because it is constant with respect to .
Step 6.1.2.2
Evaluate the limit of by plugging in for .
Step 6.1.2.3
Multiply by .
Step 6.1.3
Evaluate the limit of the denominator.
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Step 6.1.3.1
Evaluate the limit.
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Step 6.1.3.1.1
Move the limit inside the trig function because sine is continuous.
Step 6.1.3.1.2
Move the term outside of the limit because it is constant with respect to .
Step 6.1.3.2
Evaluate the limit of by plugging in for .
Step 6.1.3.3
Simplify the answer.
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Step 6.1.3.3.1
Multiply by .
Step 6.1.3.3.2
The exact value of is .
Step 6.1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
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
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.3
Differentiate using the Power Rule which states that is where .
Step 6.3.4
Multiply by .
Step 6.3.5
Differentiate using the chain rule, which states that is where and .
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Step 6.3.5.1
To apply the Chain Rule, set as .
Step 6.3.5.2
The derivative of with respect to is .
Step 6.3.5.3
Replace all occurrences of with .
Step 6.3.6
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.7
Differentiate using the Power Rule which states that is where .
Step 6.3.8
Multiply by .
Step 6.3.9
Move to the left of .
Step 6.4
Cancel the common factor of .
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Step 6.4.1
Cancel the common factor.
Step 6.4.2
Rewrite the expression.
Step 6.5
Convert from to .
Step 6.6
Evaluate the limit.
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Step 6.6.1
Move the limit inside the trig function because secant is continuous.
Step 6.6.2
Move the term outside of the limit because it is constant with respect to .
Step 6.7
Evaluate the limit of by plugging in for .
Step 6.8
Simplify the answer.
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Step 6.8.1
Multiply by .
Step 6.8.2
The exact value of is .
Step 7
Cancel the common factor of .
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Step 7.1
Cancel the common factor.
Step 7.2
Rewrite the expression.
Step 8
Evaluate the limit of which is constant as approaches .
Step 9
Simplify the answer.
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Step 9.1
Multiply by .
Step 9.2
Multiply by .
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form: