Calculus Examples

$lim_{x \to 8} \frac{100 x^{2} - 5}{5 x^{2} - 100}$

Step 1

Cancel the common factor of $100 x^{2} - 5$ and $5 x^{2} - 100$ .

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Step 1.1

Factor $5$ out of $100 x^{2}$ .

$lim_{x \to 8} \frac{5 (20 x^{2}) - 5}{5 x^{2} - 100}$

Step 1.2

Factor $5$ out of $- 5$ .

$lim_{x \to 8} \frac{5 (20 x^{2}) + 5 \cdot - 1}{5 x^{2} - 100}$

Step 1.3

Factor $5$ out of $5 (20 x^{2}) + 5 (- 1)$ .

$lim_{x \to 8} \frac{5 (20 x^{2} - 1)}{5 x^{2} - 100}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $5$ out of $5 x^{2}$ .

$lim_{x \to 8} \frac{5 (20 x^{2} - 1)}{5 (x^{2}) - 100}$

Step 1.4.2

Factor $5$ out of $- 100$ .

$lim_{x \to 8} \frac{5 (20 x^{2} - 1)}{5 x^{2} + 5 \cdot - 20}$

Step 1.4.3

Factor $5$ out of $5 x^{2} + 5 \cdot - 20$ .

$lim_{x \to 8} \frac{5 (20 x^{2} - 1)}{5 (x^{2} - 20)}$

Step 1.4.4

Cancel the common factor.

$lim_{x \to 8} \frac{5 (20 x^{2} - 1)}{5 (x^{2} - 20)}$

Step 1.4.5

Rewrite the expression.

$lim_{x \to 8} \frac{20 x^{2} - 1}{x^{2} - 20}$

Step 2

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

$\frac{lim_{x \to 8} 20 x^{2} - 1}{lim_{x \to 8} x^{2} - 20}$

Step 3

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

$\frac{lim_{x \to 8} 20 x^{2} - lim_{x \to 8} 1}{lim_{x \to 8} x^{2} - 20}$

Step 4

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$\frac{20 lim_{x \to 8} x^{2} - lim_{x \to 8} 1}{lim_{x \to 8} x^{2} - 20}$

Step 5

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

$\frac{20 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 1}{lim_{x \to 8} x^{2} - 20}$

Step 6

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

$\frac{20 {(lim_{x \to 8} x)}^{2} - 1 \cdot 1}{lim_{x \to 8} x^{2} - 20}$

Step 7

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

$\frac{20 {(lim_{x \to 8} x)}^{2} - 1 \cdot 1}{lim_{x \to 8} x^{2} - lim_{x \to 8} 20}$

Step 8

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

$\frac{20 {(lim_{x \to 8} x)}^{2} - 1 \cdot 1}{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 20}$

Step 9

Evaluate the limit of $20$ which is constant as $x$ approaches $8$ .

$\frac{20 {(lim_{x \to 8} x)}^{2} - 1 \cdot 1}{{(lim_{x \to 8} x)}^{2} - 1 \cdot 20}$

Step 10

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

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Step 10.1

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

$\frac{20 \cdot 8^{2} - 1 \cdot 1}{{(lim_{x \to 8} x)}^{2} - 1 \cdot 20}$

Step 10.2

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

$\frac{20 \cdot 8^{2} - 1 \cdot 1}{8^{2} - 1 \cdot 20}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{20 \cdot 64 - 1 \cdot 1}{8^{2} - 1 \cdot 20}$

Step 11.1.2

Multiply $20$ by $64$ .

$\frac{1280 - 1 \cdot 1}{8^{2} - 1 \cdot 20}$

Step 11.1.3

Multiply $- 1$ by $1$ .

$\frac{1280 - 1}{8^{2} - 1 \cdot 20}$

Step 11.1.4

Subtract $1$ from $1280$ .

$\frac{1279}{8^{2} - 1 \cdot 20}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

$\frac{1279}{64 - 1 \cdot 20}$

Step 11.2.2

Multiply $- 1$ by $20$ .

$\frac{1279}{64 - 20}$

Step 11.2.3

Subtract $20$ from $64$ .

$\frac{1279}{44}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1279}{44}$

Decimal Form:

$29.06\overline{81}$