Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{8 \cos (x)}{\cot (x)}$

Step 1

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$8 lim_{x \to \frac{π}{2}} \frac{\cos (x)}{\cot (x)}$

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.

$8 \frac{lim_{x \to \frac{π}{2}} \cos (x)}{lim_{x \to \frac{π}{2}} \cot (x)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Move the limit inside the trig function because cosine is continuous.

$8 \frac{\cos (lim_{x \to \frac{π}{2}} x)}{lim_{x \to \frac{π}{2}} \cot (x)}$

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$8 \frac{\cos (\frac{π}{2})}{lim_{x \to \frac{π}{2}} \cot (x)}$

Step 2.1.2.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$8 \frac{0}{lim_{x \to \frac{π}{2}} \cot (x)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Move the limit inside the trig function because cotangent is continuous.

$8 \frac{0}{\cot (lim_{x \to \frac{π}{2}} x)}$

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$8 \frac{0}{\cot (\frac{π}{2})}$

Step 2.1.3.3

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$8 (\frac{0}{0})$

Step 2.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$8 (\frac{0}{0})$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$8 (\frac{0}{0})$

Step 2.2

Since $\frac{0}{0}$ 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.

$lim_{x \to \frac{π}{2}} \frac{\cos (x)}{\cot (x)} = lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [\cot (x)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$8 lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [\cot (x)]}$

Step 2.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$8 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{\frac{d}{d x} [\cot (x)]}$

Step 2.3.3

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$8 lim_{x \to \frac{π}{2}} \frac{- \sin (x)}{- \csc^{2} (x)}$

Step 2.4

Dividing two negative values results in a positive value.

$8 lim_{x \to \frac{π}{2}} \frac{\sin (x)}{\csc^{2} (x)}$

Step 3

Evaluate the limit.

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Step 3.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$8 \frac{lim_{x \to \frac{π}{2}} \sin (x)}{lim_{x \to \frac{π}{2}} \csc^{2} (x)}$

Step 3.2

Move the limit inside the trig function because sine is continuous.

$8 \frac{\sin (lim_{x \to \frac{π}{2}} x)}{lim_{x \to \frac{π}{2}} \csc^{2} (x)}$

Step 3.3

Move the exponent $2$ from $\csc^{2} (x)$ outside the limit using the Limits Power Rule.

$8 \frac{\sin (lim_{x \to \frac{π}{2}} x)}{{(lim_{x \to \frac{π}{2}} \csc (x))}^{2}}$

Step 3.4

Move the limit inside the trig function because cosecant is continuous.

$8 \frac{\sin (lim_{x \to \frac{π}{2}} x)}{\csc^{2} (lim_{x \to \frac{π}{2}} x)}$

Step 4

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$8 \frac{\sin (\frac{π}{2})}{\csc^{2} (lim_{x \to \frac{π}{2}} x)}$

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$8 \frac{\sin (\frac{π}{2})}{\csc^{2} (\frac{π}{2})}$

Step 5

Simplify the answer.

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Step 5.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$8 \frac{1}{\csc^{2} (\frac{π}{2})}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$8 \frac{1}{1^{2}}$

Step 5.2.2

One to any power is one.

$8 (\frac{1}{1})$

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

$8 (\frac{1}{1})$

Step 5.3.2

Rewrite the expression.

$8 \cdot 1$

Step 5.4

Multiply $8$ by $1$ .

$8$