Calculus Examples

$lim_{x \to 3} \frac{9 - x^{2}}{\cos (\frac{π}{2} x)}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 3} 9 - x^{2}}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 9 - lim_{x \to 3} x^{2}}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.1.2

Evaluate the limit of $9$ which is constant as $x$ approaches $3$ .

$\frac{9 - lim_{x \to 3} x^{2}}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.1.3

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

$\frac{9 - {(lim_{x \to 3} x)}^{2}}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{9 - 3^{2}}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Raise $3$ to the power of $2$ .

$\frac{9 - 1 \cdot 9}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.3.1.2

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.2.3.2

Subtract $9$ from $9$ .

$\frac{0}{lim_{x \to 3} \cos (\frac{π}{2} x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

$\frac{0}{\cos (lim_{x \to 3} \frac{π}{2} x)}$

Step 1.3.1.2

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\cos (\frac{π}{2} lim_{x \to 3} x)}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{0}{\cos (\frac{π}{2} \cdot 3)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Combine $\frac{π}{2}$ and $3$ .

$\frac{0}{\cos (\frac{π \cdot 3}{2})}$

Step 1.3.3.2

Move $3$ to the left of $π$ .

$\frac{0}{\cos (\frac{3 \cdot π}{2})}$

Step 1.3.3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{0}{\cos (\frac{π}{2})}$

Step 1.3.3.4

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

$\frac{0}{0}$

Step 1.3.3.5

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 3} \frac{9 - x^{2}}{\cos (\frac{π}{2} x)} = lim_{x \to 3} \frac{\frac{d}{d x} [9 - x^{2}]}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [9 - x^{2}]}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.2

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{0 + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.4

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 3} \frac{0 - \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 3} \frac{0 - (2 x)}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.4.3

Multiply $2$ by $- 1$ .

$lim_{x \to 3} \frac{0 - 2 x}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.5

Subtract $2 x$ from $0$ .

$lim_{x \to 3} \frac{- 2 x}{\frac{d}{d x} [\cos (\frac{π}{2} x)]}$

Step 3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{π}{2} x$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $\frac{π}{2} x$ .

$lim_{x \to 3} \frac{- 2 x}{\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{π}{2} x]}$

Step 3.6.2

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

$lim_{x \to 3} \frac{- 2 x}{- \sin (u) \frac{d}{d x} [\frac{π}{2} x]}$

Step 3.6.3

Replace all occurrences of $u$ with $\frac{π}{2} x$ .

$lim_{x \to 3} \frac{- 2 x}{- \sin (\frac{π}{2} x) \frac{d}{d x} [\frac{π}{2} x]}$

Step 3.7

Combine $\frac{π}{2}$ and $x$ .

$lim_{x \to 3} \frac{- 2 x}{- \sin (\frac{π x}{2}) \frac{d}{d x} [\frac{π}{2} x]}$

Step 3.8

Combine $\frac{π}{2}$ and $x$ .

$lim_{x \to 3} \frac{- 2 x}{- \sin (\frac{π x}{2}) \frac{d}{d x} [\frac{π x}{2}]}$

Step 3.9

Since $\frac{π}{2}$ is constant with respect to $x$ , the derivative of $\frac{π x}{2}$ with respect to $x$ is $\frac{π}{2} \frac{d}{d x} [x]$ .

$lim_{x \to 3} \frac{- 2 x}{- \sin (\frac{π x}{2}) (\frac{π}{2} \frac{d}{d x} [x])}$

Step 3.10

Combine $\frac{π}{2}$ and $\sin (\frac{π x}{2})$ .

$lim_{x \to 3} \frac{- 2 x}{- \frac{π \sin (\frac{π x}{2})}{2} \frac{d}{d x} [x]}$

Step 3.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 3} \frac{- 2 x}{- \frac{π \sin (\frac{π x}{2})}{2} \cdot 1}$

Step 3.12

Multiply $- 1$ by $1$ .

$lim_{x \to 3} \frac{- 2 x}{- \frac{π \sin (\frac{π x}{2})}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 3} - 2 x (- \frac{2}{π \sin (\frac{π x}{2})})$

Step 5

Combine factors.

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Step 5.1

Multiply $- 1$ by $- 2$ .

$lim_{x \to 3} 2 x \frac{2}{π \sin (\frac{π x}{2})}$

Step 5.2

Combine $2$ and $\frac{2}{π \sin (\frac{π x}{2})}$ .

$lim_{x \to 3} x \frac{2 \cdot 2}{π \sin (\frac{π x}{2})}$

Step 5.3

Multiply $2$ by $2$ .

$lim_{x \to 3} x \frac{4}{π \sin (\frac{π x}{2})}$

Step 5.4

Combine $x$ and $\frac{4}{π \sin (\frac{π x}{2})}$ .

$lim_{x \to 3} \frac{x \cdot 4}{π \sin (\frac{π x}{2})}$

Step 6

Move the term $\frac{4}{π}$ outside of the limit because it is constant with respect to $x$ .

$\frac{4}{π} lim_{x \to 3} \frac{x}{\sin (\frac{π x}{2})}$

Step 7

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{4}{π} \cdot \frac{lim_{x \to 3} x}{lim_{x \to 3} \sin (\frac{π x}{2})}$

Step 8

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

$\frac{4}{π} \cdot \frac{lim_{x \to 3} x}{\sin (lim_{x \to 3} \frac{π x}{2})}$

Step 9

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{4}{π} \cdot \frac{lim_{x \to 3} x}{\sin (\frac{π}{2} lim_{x \to 3} x)}$

Step 10

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{4}{π} \cdot \frac{3}{\sin (\frac{π}{2} lim_{x \to 3} x)}$

Step 10.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{4}{π} \cdot \frac{3}{\sin (\frac{π}{2} \cdot 3)}$

Step 11

Simplify the answer.

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Step 11.1

Separate fractions.

$\frac{4}{π} (\frac{3}{1} \cdot \frac{1}{\sin (\frac{π}{2} \cdot 3)})$

Step 11.2

Convert from $\frac{1}{\sin (\frac{π}{2} \cdot 3)}$ to $\csc (\frac{π}{2} \cdot 3)$ .

$\frac{4}{π} (\frac{3}{1} \csc (\frac{π}{2} \cdot 3))$

Step 11.3

Divide $3$ by $1$ .

$\frac{4}{π} (3 \csc (\frac{π}{2} \cdot 3))$

Step 11.4

Combine $\frac{π}{2}$ and $3$ .

$\frac{4}{π} (3 \csc (\frac{π \cdot 3}{2}))$

Step 11.5

Move $3$ to the left of $π$ .

$\frac{4}{π} (3 \csc (\frac{3 \cdot π}{2}))$

Step 11.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$\frac{4}{π} (3 (- \csc (\frac{π}{2})))$

Step 11.7

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

$\frac{4}{π} (3 (- 1 \cdot 1))$

Step 11.8

Multiply $3 (- 1 \cdot 1)$ .

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Step 11.8.1

Multiply $- 1$ by $1$ .

$\frac{4}{π} (3 \cdot - 1)$

Step 11.8.2

Multiply $3$ by $- 1$ .

$\frac{4}{π} \cdot - 3$

Step 11.9

Multiply $\frac{4}{π} \cdot - 3$ .

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Step 11.9.1

Combine $\frac{4}{π}$ and $- 3$ .

$\frac{4 \cdot - 3}{π}$

Step 11.9.2

Multiply $4$ by $- 3$ .

$\frac{- 12}{π}$

Step 11.10

Move the negative in front of the fraction.

$- \frac{12}{π}$