Calculus Examples

$lim_{x \to - 8} {(\frac{x - 7}{x - 5})}^{2}$

Step 1

Move the exponent $2$ from ${(\frac{x - 7}{x - 5})}^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - 8} \frac{x - 7}{x - 5})}^{2}$

Step 2

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

${(\frac{lim_{x \to - 8} x - 7}{lim_{x \to - 8} x - 5})}^{2}$

Step 3

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

${(\frac{lim_{x \to - 8} x - lim_{x \to - 8} 7}{lim_{x \to - 8} x - 5})}^{2}$

Step 4

Evaluate the limit of $7$ which is constant as $x$ approaches $- 8$ .

${(\frac{lim_{x \to - 8} x - 1 \cdot 7}{lim_{x \to - 8} x - 5})}^{2}$

Step 5

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

${(\frac{lim_{x \to - 8} x - 1 \cdot 7}{lim_{x \to - 8} x - lim_{x \to - 8} 5})}^{2}$

Step 6

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

${(\frac{lim_{x \to - 8} x - 1 \cdot 7}{lim_{x \to - 8} x - 1 \cdot 5})}^{2}$

Step 7

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

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Step 7.1

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

${(\frac{- 8 - 1 \cdot 7}{lim_{x \to - 8} x - 1 \cdot 5})}^{2}$

Step 7.2

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

${(\frac{- 8 - 1 \cdot 7}{- 8 - 1 \cdot 5})}^{2}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 1$ by $7$ .

${(\frac{- 8 - 7}{- 8 - 1 \cdot 5})}^{2}$

Step 8.1.2

Subtract $7$ from $- 8$ .

${(\frac{- 15}{- 8 - 1 \cdot 5})}^{2}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $5$ .

${(\frac{- 15}{- 8 - 5})}^{2}$

Step 8.2.2

Subtract $5$ from $- 8$ .

${(\frac{- 15}{- 13})}^{2}$

Step 8.3

Dividing two negative values results in a positive value.

${(\frac{15}{13})}^{2}$

Step 8.4

Apply the product rule to $\frac{15}{13}$ .

$\frac{15^{2}}{13^{2}}$

Step 8.5

Raise $15$ to the power of $2$ .

$\frac{225}{13^{2}}$

Step 8.6

Raise $13$ to the power of $2$ .

$\frac{225}{169}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{225}{169}$

Decimal Form:

$1.33136094 \dots$