Calculus Examples

$\frac{lim_{x \to 8} 10^{8} x^{5} + 10^{6} x^{4} + 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 1

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

$\frac{lim_{x \to 8} 10^{8} x^{5} + lim_{x \to 8} 10^{6} x^{4} + lim_{x \to 8} 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 2

Move the term $10^{8}$ outside of the limit because it is constant with respect to $x$ .

$\frac{10^{8} lim_{x \to 8} x^{5} + lim_{x \to 8} 10^{6} x^{4} + lim_{x \to 8} 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{10^{8} {(lim_{x \to 8} x)}^{5} + lim_{x \to 8} 10^{6} x^{4} + lim_{x \to 8} 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 4

Move the term $10^{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{10^{8} {(lim_{x \to 8} x)}^{5} + 10^{6} lim_{x \to 8} x^{4} + lim_{x \to 8} 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{10^{8} {(lim_{x \to 8} x)}^{5} + 10^{6} {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 10^{4} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 6

Move the term $10^{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{10^{8} {(lim_{x \to 8} x)}^{5} + 10^{6} {(lim_{x \to 8} x)}^{4} + 10^{4} lim_{x \to 8} x^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 7

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

$\frac{10^{8} {(lim_{x \to 8} x)}^{5} + 10^{6} {(lim_{x \to 8} x)}^{4} + 10^{4} {(lim_{x \to 8} x)}^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

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 $x$ .

$\frac{10^{8} \cdot 8^{5} + 10^{6} {(lim_{x \to 8} x)}^{4} + 10^{4} {(lim_{x \to 8} x)}^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 8.2

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

$\frac{10^{8} \cdot 8^{5} + 10^{6} \cdot 8^{4} + 10^{4} {(lim_{x \to 8} x)}^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 8.3

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

$\frac{10^{8} \cdot 8^{5} + 10^{6} \cdot 8^{4} + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $10$ to the power of $8$ .

$\frac{100000000 \cdot 8^{5} + 10^{6} \cdot 8^{4} + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.2

Raise $8$ to the power of $5$ .

$\frac{100000000 \cdot 32768 + 10^{6} \cdot 8^{4} + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.3

Multiply $100000000$ by $32768$ .

$\frac{3276800000000 + 10^{6} \cdot 8^{4} + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.4

Raise $10$ to the power of $6$ .

$\frac{3276800000000 + 1000000 \cdot 8^{4} + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.5

Raise $8$ to the power of $4$ .

$\frac{3276800000000 + 1000000 \cdot 4096 + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.6

Multiply $1000000$ by $4096$ .

$\frac{3276800000000 + 4096000000 + 10^{4} \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.7

Raise $10$ to the power of $4$ .

$\frac{3276800000000 + 4096000000 + 10000 \cdot 8^{2}}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.8

Raise $8$ to the power of $2$ .

$\frac{3276800000000 + 4096000000 + 10000 \cdot 64}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.9

Multiply $10000$ by $64$ .

$\frac{3276800000000 + 4096000000 + 640000}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.10

Add $3276800000000$ and $4096000000$ .

$\frac{3280896000000 + 640000}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.1.11

Add $3280896000000$ and $640000$ .

$\frac{3280896640000}{10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Factor $10^{5} x^{3}$ out of $10^{9} x^{6} + 10^{7} x^{5} + 10^{5} x^{3}$ .

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Step 9.2.1.1

Factor $10^{5} x^{3}$ out of $10^{9} x^{6}$ .

$\frac{3280896640000}{10^{5} x^{3} (10^{4} x^{3}) + 10^{7} x^{5} + 10^{5} x^{3}}$

Step 9.2.1.2

Factor $10^{5} x^{3}$ out of $10^{7} x^{5}$ .

$\frac{3280896640000}{10^{5} x^{3} (10^{4} x^{3}) + 10^{5} x^{3} (10^{2} x^{2}) + 10^{5} x^{3}}$

Step 9.2.1.3

Factor $10^{5} x^{3}$ out of $10^{5} x^{3}$ .

$\frac{3280896640000}{10^{5} x^{3} (10^{4} x^{3}) + 10^{5} x^{3} (10^{2} x^{2}) + 10^{5} x^{3} (1)}$

Step 9.2.1.4

Factor $10^{5} x^{3}$ out of $10^{5} x^{3} (10^{4} x^{3}) + 10^{5} x^{3} (10^{2} x^{2})$ .

$\frac{3280896640000}{10^{5} x^{3} (10^{4} x^{3} + 10^{2} x^{2}) + 10^{5} x^{3} (1)}$

Step 9.2.1.5

Factor $10^{5} x^{3}$ out of $10^{5} x^{3} (10^{4} x^{3} + 10^{2} x^{2}) + 10^{5} x^{3} (1)$ .

$\frac{3280896640000}{10^{5} x^{3} (10^{4} x^{3} + 10^{2} x^{2} + 1)}$

Step 9.2.2

Raise $10$ to the power of $4$ .

$\frac{3280896640000}{10^{5} x^{3} (10000 x^{3} + 10^{2} x^{2} + 1)}$

Step 9.2.3

Raise $10$ to the power of $2$ .

$\frac{3280896640000}{10^{5} x^{3} (10000 x^{3} + 100 x^{2} + 1)}$

Step 9.2.4

Factor $10000 x^{3} + 100 x^{2} + 1$ using the rational roots test.

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Step 9.2.4.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 10000, \pm 2, \pm 5000, \pm 4, \pm 2500, \pm 5, \pm 2000, \pm 8, \pm 1250, \pm 10, \pm 1000, \pm 16, \pm 625, \pm 20, \pm 500, \pm 25, \pm 400, \pm 40, \pm 250, \pm 50, \pm 200, \pm 80, \pm 125, \pm 100$

Step 9.2.4.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.0001, \pm 0.5, \pm 0.0002, \pm 0.25, \pm 0.0004, \pm 0.2, \pm 0.0005, \pm 0.125, \pm 0.0008, \pm 0.1, \pm 0.001, \pm 0.0625, \pm 0.0016, \pm 0.05, \pm 0.002, \pm 0.04, \pm 0.0025, \pm 0.025, \pm 0.004, \pm 0.02, \pm 0.005, \pm 0.0125, \pm 0.008, \pm 0.01$

Step 9.2.4.3

Substitute $- 0.05$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.05$ is a root of the polynomial.

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Step 9.2.4.3.1

Substitute $- 0.05$ into the polynomial.

$10000 {(- 0.05)}^{3} + 100 {(- 0.05)}^{2} + 1$

Step 9.2.4.3.2

Raise $- 0.05$ to the power of $3$ .

$10000 \cdot - 0.000125 + 100 {(- 0.05)}^{2} + 1$

Step 9.2.4.3.3

Multiply $10000$ by $- 0.000125$ .

$- 1.25 + 100 {(- 0.05)}^{2} + 1$

Step 9.2.4.3.4

Raise $- 0.05$ to the power of $2$ .

$- 1.25 + 100 \cdot 0.0025 + 1$

Step 9.2.4.3.5

Multiply $100$ by $0.0025$ .

$- 1.25 + 0.25 + 1$

Step 9.2.4.3.6

Add $- 1.25$ and $0.25$ .

$- 1 + 1$

Step 9.2.4.3.7

Add $- 1$ and $1$ .

$0$

Step 9.2.4.4

Since $- 0.05$ is a known root, divide the polynomial by $20 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{10000 x^{3} + 100 x^{2} + 1}{20 x + 1}$

Step 9.2.4.5

Divide $10000 x^{3} + 100 x^{2} + 1$ by $20 x + 1$ .

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Step 9.2.4.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$20 x$

$1$

$10000 x^{3}$

$100 x^{2}$

$0 x$

$1$

Step 9.2.4.5.2

Divide the highest order term in the dividend $10000 x^{3}$ by the highest order term in divisor $20 x$ .

					$500 x^{2}$
	$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$

Step 9.2.4.5.3

Multiply the new quotient term by the divisor.

				$500 x^{2}$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			+	$10000 x^{3}$	+	$500 x^{2}$

Step 9.2.4.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $10000 x^{3} + 500 x^{2}$

				$500 x^{2}$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$

Step 9.2.4.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$500 x^{2}$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$

Step 9.2.4.5.6

Pull the next terms from the original dividend down into the current dividend.

				$500 x^{2}$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$

Step 9.2.4.5.7

Divide the highest order term in the dividend $- 400 x^{2}$ by the highest order term in divisor $20 x$ .

				$500 x^{2}$	-	$20 x$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$

Step 9.2.4.5.8

Multiply the new quotient term by the divisor.

				$500 x^{2}$	-	$20 x$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					-	$400 x^{2}$	-	$20 x$

Step 9.2.4.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 400 x^{2} - 20 x$

				$500 x^{2}$	-	$20 x$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$

Step 9.2.4.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$500 x^{2}$	-	$20 x$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$

Step 9.2.4.5.11

Pull the next terms from the original dividend down into the current dividend.

				$500 x^{2}$	-	$20 x$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$	+	$1$

Step 9.2.4.5.12

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $20 x$ .

				$500 x^{2}$	-	$20 x$	+	$1$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$	+	$1$

Step 9.2.4.5.13

Multiply the new quotient term by the divisor.

				$500 x^{2}$	-	$20 x$	+	$1$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$	+	$1$
							+	$20 x$	+	$1$

Step 9.2.4.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 1$

				$500 x^{2}$	-	$20 x$	+	$1$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$	+	$1$
							-	$20 x$	-	$1$

Step 9.2.4.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$500 x^{2}$	-	$20 x$	+	$1$
$20 x$	+	$1$		$10000 x^{3}$	+	$100 x^{2}$	+	$0 x$	+	$1$
			-	$10000 x^{3}$	-	$500 x^{2}$
					-	$400 x^{2}$	+	$0 x$
					+	$400 x^{2}$	+	$20 x$
							+	$20 x$	+	$1$
							-	$20 x$	-	$1$
										$0$

Step 9.2.4.5.16

Since the remander is $0$ , the final answer is the quotient.

$500 x^{2} - 20 x + 1$

Step 9.2.4.6

Write $10000 x^{3} + 100 x^{2} + 1$ as a set of factors.

$\frac{3280896640000}{10^{5} x^{3} ((20 x + 1) (500 x^{2} - 20 x + 1))}$

$\frac{3280896640000}{10^{5} x^{3} (20 x + 1) (500 x^{2} - 20 x + 1)}$

Step 9.2.5

Raise $10$ to the power of $5$ .

$\frac{3280896640000}{100000 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1)}$

Step 9.3

Cancel the common factor of $3280896640000$ and $100000$ .

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Step 9.3.1

Factor $20000$ out of $3280896640000$ .

$\frac{20000 (164044832)}{100000 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1)}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $20000$ out of $100000 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1)$ .

$\frac{20000 (164044832)}{20000 (5 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1))}$

Step 9.3.2.2

Cancel the common factor.

$\frac{20000 \cdot 164044832}{20000 (5 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1))}$

Step 9.3.2.3

Rewrite the expression.

$\frac{164044832}{5 x^{3} (20 x + 1) (500 x^{2} - 20 x + 1)}$