Calculus Examples

$lim_{x \to π} \frac{8 \cos (x) + 8}{{(x - π)}^{2}}$

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.

$\frac{lim_{x \to π} 8 \cos (x) + 8}{lim_{x \to π} {(x - π)}^{2}}$

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 $π$ .

$\frac{lim_{x \to π} 8 \cos (x) + lim_{x \to π} 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.1.2

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

$\frac{8 lim_{x \to π} \cos (x) + lim_{x \to π} 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.1.3

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

$\frac{8 \cos (lim_{x \to π} x) + lim_{x \to π} 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.1.4

Evaluate the limit of $8$ which is constant as $x$ approaches $π$ .

$\frac{8 \cos (lim_{x \to π} x) + 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.2

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

$\frac{8 \cos (π) + 8}{lim_{x \to π} {(x - π)}^{2}}$

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

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

$\frac{8 (- \cos (0)) + 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.3.1.2

The exact value of $\cos (0)$ is $1$ .

$\frac{8 (- 1 \cdot 1) + 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.3.1.3

Multiply $8 (- 1 \cdot 1)$ .

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Step 1.2.3.1.3.1

Multiply $- 1$ by $1$ .

$\frac{8 \cdot - 1 + 8}{lim_{x \to π} {(x - π)}^{2}}$

Step 1.2.3.1.3.2

Multiply $8$ by $- 1$ .

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

Step 1.2.3.2

Add $- 8$ and $8$ .

$\frac{0}{lim_{x \to π} {(x - π)}^{2}}$

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 exponent $2$ from ${(x - π)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to π} x - π)}^{2}}$

Step 1.3.1.2

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

$\frac{0}{{(lim_{x \to π} x - lim_{x \to π} π)}^{2}}$

Step 1.3.1.3

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

$\frac{0}{{(lim_{x \to π} x - π)}^{2}}$

Step 1.3.2

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

$\frac{0}{{(π - π)}^{2}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Subtract $π$ from $π$ .

$\frac{0}{0^{2}}$

Step 1.3.3.2

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.3.3.3

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.

$lim_{x \to π} \frac{8 \cos (x) + 8}{{(x - π)}^{2}} = lim_{x \to π} \frac{\frac{d}{d x} [8 \cos (x) + 8]}{\frac{d}{d x} [{(x - π)}^{2}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of $8 \cos (x) + 8$ with respect to $x$ is $\frac{d}{d x} [8 \cos (x)] + \frac{d}{d x} [8]$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [8 \cos (x)]$ .

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Step 3.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \cos (x)$ with respect to $x$ is $8 \frac{d}{d x} [\cos (x)]$ .

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $- 1$ by $8$ .

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

Step 3.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to π} \frac{- 8 \sin (x) + 0}{\frac{d}{d x} [{(x - π)}^{2}]}$

Step 3.5

Add $- 8 \sin (x)$ and $0$ .

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

Step 3.6

Rewrite ${(x - π)}^{2}$ as $(x - π) (x - π)$ .

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

Step 3.7

Expand $(x - π) (x - π)$ using the FOIL Method.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Apply the distributive property.

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

Step 3.7.3

Apply the distributive property.

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

Step 3.8

Simplify and combine like terms.

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Step 3.8.1

Simplify each term.

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Step 3.8.1.1

Multiply $x$ by $x$ .

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

Step 3.8.1.2

Multiply $- π (- π)$ .

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Step 3.8.1.2.1

Multiply $- 1$ by $- 1$ .

$lim_{x \to π} \frac{- 8 \sin (x)}{\frac{d}{d x} [x^{2} + x (- π) - π x + 1 π π]}$

Step 3.8.1.2.2

Multiply $π$ by $1$ .

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

Step 3.8.1.2.3

Raise $π$ to the power of $1$ .

$lim_{x \to π} \frac{- 8 \sin (x)}{\frac{d}{d x} [x^{2} + x (- π) - π x + π^{1} π]}$

Step 3.8.1.2.4

Raise $π$ to the power of $1$ .

$lim_{x \to π} \frac{- 8 \sin (x)}{\frac{d}{d x} [x^{2} + x (- π) - π x + π^{1} π^{1}]}$

Step 3.8.1.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to π} \frac{- 8 \sin (x)}{\frac{d}{d x} [x^{2} + x (- π) - π x + π^{1 + 1}]}$

Step 3.8.1.2.6

Add $1$ and $1$ .

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

Step 3.8.2

Subtract $π x$ from $x (- π)$ .

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Step 3.8.2.1

Move $x$ .

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

Step 3.8.2.2

Subtract $π x$ from $- π x$ .

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

$lim_{x \to π} \frac{- 8 \sin (x)}{2 x - 2 π \cdot 1 + \frac{d}{d x} [π^{2}]}$

Step 3.13

Multiply $- 2$ by $1$ .

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

Step 3.14

Since $π^{2}$ is constant with respect to $x$ , the derivative of $π^{2}$ with respect to $x$ is $0$ .

$lim_{x \to π} \frac{- 8 \sin (x)}{2 x - 2 π + 0}$

Step 3.15

Add $2 x - 2 π$ and $0$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $- 8$ and $2 x - 2 π$ .

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Step 4.1.1

Factor $2$ out of $- 8 \sin (x)$ .

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 x$ .

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

Step 4.1.2.2

Factor $2$ out of $- 2 π$ .

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

Step 4.1.2.3

Factor $2$ out of $2 (x) + 2 (- π)$ .

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

Step 4.1.2.4

Cancel the common factor.

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

Step 4.1.2.5

Rewrite the expression.

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

Step 4.2

Move the term $- 4$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

Apply L'Hospital's rule.

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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.

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

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Simplify the answer.

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Step 5.1.2.3.1

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

$- 4 \frac{\sin (0)}{lim_{x \to π} x - π}$

Step 5.1.2.3.2

The exact value of $\sin (0)$ is $0$ .

$- 4 \frac{0}{lim_{x \to π} x - π}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Evaluate the limit.

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Step 5.1.3.1.1

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

$- 4 \frac{0}{lim_{x \to π} x - lim_{x \to π} π}$

Step 5.1.3.1.2

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

$- 4 \frac{0}{lim_{x \to π} x - π}$

Step 5.1.3.2

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

$- 4 \frac{0}{π - π}$

Step 5.1.3.3

Subtract $π$ from $π$ .

$- 4 (\frac{0}{0})$

Step 5.1.3.4

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

Undefined

$- 4 (\frac{0}{0})$

Step 5.1.4

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

Undefined

$- 4 (\frac{0}{0})$

Step 5.2

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.

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

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$- 4 lim_{x \to π} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x - π]}$

Step 5.3.2

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

$- 4 lim_{x \to π} \frac{\cos (x)}{\frac{d}{d x} [x - π]}$

Step 5.3.3

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

$- 4 lim_{x \to π} \frac{\cos (x)}{\frac{d}{d x} [x] + \frac{d}{d x} [- π]}$

Step 5.3.4

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

$- 4 lim_{x \to π} \frac{\cos (x)}{1 + \frac{d}{d x} [- π]}$

Step 5.3.5

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

$- 4 lim_{x \to π} \frac{\cos (x)}{1 + 0}$

Step 5.3.6

Add $1$ and $0$ .

$- 4 lim_{x \to π} \frac{\cos (x)}{1}$

Step 5.4

Divide $\cos (x)$ by $1$ .

$- 4 lim_{x \to π} \cos (x)$

Step 6

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

$- 4 \cos (lim_{x \to π} x)$

Step 7

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

$- 4 \cos (π)$

Step 8

Simplify the answer.

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Step 8.1

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

$- 4 (- \cos (0))$

Step 8.2

The exact value of $\cos (0)$ is $1$ .

$- 4 (- 1 \cdot 1)$

Step 8.3

Multiply $- 4 (- 1 \cdot 1)$ .

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Step 8.3.1

Multiply $- 1$ by $1$ .

$- 4 \cdot - 1$

Step 8.3.2

Multiply $- 4$ by $- 1$ .

$4$