Calculus Examples

$lim_{x \to - 5} \frac{5 x - 10}{x^{2 - 25}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 5$ .

$\frac{lim_{x \to - 5} 5 x - 10}{lim_{x \to - 5} x^{2 - 25}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{lim_{x \to - 5} 5 x - lim_{x \to - 5} 10}{lim_{x \to - 5} x^{2 - 25}}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to - 5} x - lim_{x \to - 5} 10}{lim_{x \to - 5} x^{2 - 25}}$

Step 4

Evaluate the limit of $10$ which is constant as $x$ approaches $- 5$ .

$\frac{5 lim_{x \to - 5} x - 1 \cdot 10}{lim_{x \to - 5} x^{2 - 25}}$

Step 5

Move the exponent $2 - 25$ from $x^{2 - 25}$ outside the limit using the Limits Power Rule.

$\frac{5 lim_{x \to - 5} x - 1 \cdot 10}{{(lim_{x \to - 5} x)}^{2 - 25}}$

Step 6

Evaluate the limits by plugging in $- 5$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{5 \cdot - 5 - 1 \cdot 10}{{(lim_{x \to - 5} x)}^{2 - 25}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{5 \cdot - 5 - 1 \cdot 10}{{(- 5)}^{2 - 25}}$

Step 7

Simplify the answer.

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Step 7.1

Move ${(- 5)}^{2 - 25}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$(5 \cdot - 5 - 1 \cdot 10) {(- 5)}^{- (2 - 25)}$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $5$ by $- 5$ .

$(- 25 - 1 \cdot 10) {(- 5)}^{- (2 - 25)}$

Step 7.2.2

Multiply $- 1$ by $10$ .

$(- 25 - 10) {(- 5)}^{- (2 - 25)}$

Step 7.3

Subtract $10$ from $- 25$ .

$- 35 {(- 5)}^{- (2 - 25)}$

Step 7.4

Subtract $25$ from $2$ .

$- 35 {(- 5)}^{- - 23}$

Step 7.5

Multiply $- 1$ by $- 23$ .

$- 35 {(- 5)}^{23}$

Step 7.6

Raise $- 5$ to the power of $23$ .

$- 35 \cdot - 11920928955078100$

Step 7.7

Multiply $- 35$ by $- 11920928955078100$ .

$417232513427733500$

Step 8

The result can be shown in multiple forms.

Exact Form:

$417232513427733500$

Decimal Form:

$4.17232513 \cdot 10^{17}$