Calculus Examples

$lim_{x \to - 8} 5 + \frac{100}{x} + \frac{x^{3} \sin (4)}{x^{2}}$

Step 1

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 1.1

Factor $x^{2}$ out of $x^{3} \sin (4)$ .

$lim_{x \to - 8} 5 + \frac{100}{x} + \frac{x^{2} (x \sin (4))}{x^{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

$lim_{x \to - 8} 5 + \frac{100}{x} + \frac{x^{2} (x \sin (4))}{x^{2} \cdot 1}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to - 8} 5 + \frac{100}{x} + \frac{x^{2} (x \sin (4))}{x^{2} \cdot 1}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to - 8} 5 + \frac{100}{x} + \frac{x \sin (4)}{1}$

Step 1.2.4

Divide $x \sin (4)$ by $1$ .

$lim_{x \to - 8} 5 + \frac{100}{x} + x \sin (4)$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$lim_{x \to - 8} 5 + lim_{x \to - 8} \frac{100}{x} + lim_{x \to - 8} x \sin (4)$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $- 8$ .

$5 + lim_{x \to - 8} \frac{100}{x} + lim_{x \to - 8} x \sin (4)$

Step 4

Move the term $100$ outside of the limit because it is constant with respect to $x$ .

$5 + 100 lim_{x \to - 8} \frac{1}{x} + lim_{x \to - 8} x \sin (4)$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$5 + 100 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x} + lim_{x \to - 8} x \sin (4)$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$5 + 100 \frac{1}{lim_{x \to - 8} x} + lim_{x \to - 8} x \sin (4)$

Step 7

Move the term $\sin (4)$ outside of the limit because it is constant with respect to $x$ .

$5 + 100 \frac{1}{lim_{x \to - 8} x} + \sin (4) lim_{x \to - 8} x$

Step 8

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$5 + 100 (\frac{1}{- 8}) + \sin (4) lim_{x \to - 8} x$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$5 + 100 (\frac{1}{- 8}) + \sin (4) \cdot - 8$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Cancel the common factor of $4$ .

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Step 9.1.1.1

Factor $4$ out of $100$ .

$5 + 4 (25) \frac{1}{- 8} + \sin (4) \cdot - 8$

Step 9.1.1.2

Factor $4$ out of $- 8$ .

$5 + 4 \cdot 25 \frac{1}{4 \cdot - 2} + \sin (4) \cdot - 8$

Step 9.1.1.3

Cancel the common factor.

$5 + 4 \cdot 25 \frac{1}{4 \cdot - 2} + \sin (4) \cdot - 8$

Step 9.1.1.4

Rewrite the expression.

$5 + 25 (\frac{1}{- 2}) + \sin (4) \cdot - 8$

Step 9.1.2

Combine $25$ and $\frac{1}{- 2}$ .

$5 + \frac{25}{- 2} + \sin (4) \cdot - 8$

Step 9.1.3

Move the negative in front of the fraction.

$5 - \frac{25}{2} + \sin (4) \cdot - 8$

Step 9.1.4

Evaluate $\sin (4)$ .

$5 - \frac{25}{2} + 0.06975647 \cdot - 8$

Step 9.1.5

Multiply $0.06975647$ by $- 8$ .

$5 - \frac{25}{2} - 0.55805178$

Step 9.2

Find the common denominator.

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Step 9.2.1

Write $5$ as a fraction with denominator $1$ .

$\frac{5}{1} - \frac{25}{2} - 0.55805178$

Step 9.2.2

Multiply $\frac{5}{1}$ by $\frac{2}{2}$ .

$\frac{5}{1} \cdot \frac{2}{2} - \frac{25}{2} - 0.55805178$

Step 9.2.3

Multiply $\frac{5}{1}$ by $\frac{2}{2}$ .

$\frac{5 \cdot 2}{2} - \frac{25}{2} - 0.55805178$

Step 9.2.4

Write $- 0.55805178$ as a fraction with denominator $1$ .

$\frac{5 \cdot 2}{2} - \frac{25}{2} + \frac{- 0.55805178}{1}$

Step 9.2.5

Multiply $\frac{- 0.55805178}{1}$ by $\frac{2}{2}$ .

$\frac{5 \cdot 2}{2} - \frac{25}{2} + \frac{- 0.55805178}{1} \cdot \frac{2}{2}$

Step 9.2.6

Multiply $\frac{- 0.55805178}{1}$ by $\frac{2}{2}$ .

$\frac{5 \cdot 2}{2} - \frac{25}{2} + \frac{- 0.55805178 \cdot 2}{2}$

Step 9.3

Combine the numerators over the common denominator.

$\frac{5 \cdot 2 - 25 - 0.55805178 \cdot 2}{2}$

Step 9.4

Simplify each term.

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Step 9.4.1

Multiply $5$ by $2$ .

$\frac{10 - 25 - 0.55805178 \cdot 2}{2}$

Step 9.4.2

Multiply $- 0.55805178$ by $2$ .

$\frac{10 - 25 - 1.11610357}{2}$

Step 9.5

Subtract $25$ from $10$ .

$\frac{- 15 - 1.11610357}{2}$

Step 9.6

Subtract $1.11610357$ from $- 15$ .

$\frac{- 16.11610357}{2}$

Step 9.7

Divide $- 16.11610357$ by $2$ .

$- 8.05805178$