Calculus Examples

Evaluate the Limit limit as x approaches 0 of x/(tan(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 by plugging in for .
Step 1.1.3
Evaluate the limit of the denominator.
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Step 1.1.3.1
Move the limit inside the trig function because tangent is continuous.
Step 1.1.3.2
Evaluate the limit of by plugging in for .
Step 1.1.3.3
The exact value of is .
Step 1.1.3.4
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
Differentiate using the Power Rule which states that is where .
Step 1.3.3
The derivative of with respect to is .
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
Move the exponent from outside the limit using the Limits Power Rule.
Step 2.4
Move the limit inside the trig function because secant is continuous.
Step 3
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Rewrite as .
Step 4.2
Rewrite as .
Step 4.3
Rewrite in terms of sines and cosines.
Step 4.4
Multiply by the reciprocal of the fraction to divide by .
Step 4.5
Multiply by .
Step 4.6
The exact value of is .
Step 4.7
One to any power is one.