Calculus Examples

$\frac{lim_{x \to 8} 3 x^{3} + 2 x + 1}{100 x^{2} - 1000 x - 10000}$

Step 1

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

$\frac{lim_{x \to 8} 3 x^{3} + lim_{x \to 8} 2 x + lim_{x \to 8} 1}{100 x^{2} - 1000 x - 10000}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 8} x^{3} + lim_{x \to 8} 2 x + lim_{x \to 8} 1}{100 x^{2} - 1000 x - 10000}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 2 x + lim_{x \to 8} 1}{100 x^{2} - 1000 x - 10000}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 8} x)}^{3} + 2 lim_{x \to 8} x + lim_{x \to 8} 1}{100 x^{2} - 1000 x - 10000}$

Step 5

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

$\frac{3 {(lim_{x \to 8} x)}^{3} + 2 lim_{x \to 8} x + 1}{100 x^{2} - 1000 x - 10000}$

Step 6

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

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Step 6.1

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

$\frac{3 \cdot 8^{3} + 2 lim_{x \to 8} x + 1}{100 x^{2} - 1000 x - 10000}$

Step 6.2

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

$\frac{3 \cdot 8^{3} + 2 \cdot 8 + 1}{100 x^{2} - 1000 x - 10000}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $8$ to the power of $3$ .

$\frac{3 \cdot 512 + 2 \cdot 8 + 1}{100 x^{2} - 1000 x - 10000}$

Step 7.1.2

Multiply $3$ by $512$ .

$\frac{1536 + 2 \cdot 8 + 1}{100 x^{2} - 1000 x - 10000}$

Step 7.1.3

Multiply $2$ by $8$ .

$\frac{1536 + 16 + 1}{100 x^{2} - 1000 x - 10000}$

Step 7.1.4

Add $1536$ and $16$ .

$\frac{1552 + 1}{100 x^{2} - 1000 x - 10000}$

Step 7.1.5

Add $1552$ and $1$ .

$\frac{1553}{100 x^{2} - 1000 x - 10000}$

Step 7.2

Factor $100$ out of $100 x^{2} - 1000 x - 10000$ .

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Step 7.2.1

Factor $100$ out of $100 x^{2}$ .

$\frac{1553}{100 (x^{2}) - 1000 x - 10000}$

Step 7.2.2

Factor $100$ out of $- 1000 x$ .

$\frac{1553}{100 (x^{2}) + 100 (- 10 x) - 10000}$

Step 7.2.3

Factor $100$ out of $- 10000$ .

$\frac{1553}{100 x^{2} + 100 (- 10 x) + 100 \cdot - 100}$

Step 7.2.4

Factor $100$ out of $100 x^{2} + 100 (- 10 x)$ .

$\frac{1553}{100 (x^{2} - 10 x) + 100 \cdot - 100}$

Step 7.2.5

Factor $100$ out of $100 (x^{2} - 10 x) + 100 \cdot - 100$ .

$\frac{1553}{100 (x^{2} - 10 x - 100)}$