Calculus Examples

$lim_{x \to - 8} \frac{- 2 x + 5}{5 x - \sqrt{25 x^{2} + x - 5}}$

Step 1

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

$\frac{lim_{x \to - 8} - 2 x + 5}{lim_{x \to - 8} 5 x - \sqrt{25 x^{2} + x - 5}}$

Step 2

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

$\frac{- lim_{x \to - 8} 2 x + lim_{x \to - 8} 5}{lim_{x \to - 8} 5 x - \sqrt{25 x^{2} + x - 5}}$

Step 3

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

$\frac{- 2 lim_{x \to - 8} x + lim_{x \to - 8} 5}{lim_{x \to - 8} 5 x - \sqrt{25 x^{2} + x - 5}}$

Step 4

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

$\frac{- 2 lim_{x \to - 8} x + 5}{lim_{x \to - 8} 5 x - \sqrt{25 x^{2} + x - 5}}$

Step 5

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

$\frac{- 2 lim_{x \to - 8} x + 5}{lim_{x \to - 8} 5 x - lim_{x \to - 8} \sqrt{25 x^{2} + x - 5}}$

Step 6

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

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - lim_{x \to - 8} \sqrt{25 x^{2} + x - 5}}$

Step 7

Move the limit under the radical sign.

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - \sqrt{lim_{x \to - 8} 25 x^{2} + x - 5}}$

Step 8

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

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - \sqrt{lim_{x \to - 8} 25 x^{2} + lim_{x \to - 8} x - lim_{x \to - 8} 5}}$

Step 9

Move the term $25$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - \sqrt{25 lim_{x \to - 8} x^{2} + lim_{x \to - 8} x - lim_{x \to - 8} 5}}$

Step 10

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

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - \sqrt{25 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - lim_{x \to - 8} 5}}$

Step 11

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

$\frac{- 2 lim_{x \to - 8} x + 5}{5 lim_{x \to - 8} x - \sqrt{25 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 5}}$

Step 12

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

Tap for more steps...

Step 12.1

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

$\frac{- 2 \cdot - 8 + 5}{5 lim_{x \to - 8} x - \sqrt{25 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 5}}$

Step 12.2

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

$\frac{- 2 \cdot - 8 + 5}{5 \cdot - 8 - \sqrt{25 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} x - 1 \cdot 5}}$

Step 12.3

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

$\frac{- 2 \cdot - 8 + 5}{5 \cdot - 8 - \sqrt{25 {(- 8)}^{2} + lim_{x \to - 8} x - 1 \cdot 5}}$

Step 12.4

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

$\frac{- 2 \cdot - 8 + 5}{5 \cdot - 8 - \sqrt{25 {(- 8)}^{2} - 8 - 1 \cdot 5}}$

Step 13

Simplify the answer.

Tap for more steps...

Step 13.1

Simplify the numerator.

Tap for more steps...

Step 13.1.1

Multiply $- 2$ by $- 8$ .

$\frac{16 + 5}{5 \cdot - 8 - \sqrt{25 {(- 8)}^{2} - 8 - 1 \cdot 5}}$

Step 13.1.2

Add $16$ and $5$ .

$\frac{21}{5 \cdot - 8 - \sqrt{25 {(- 8)}^{2} - 8 - 1 \cdot 5}}$

Step 13.2

Simplify the denominator.

Tap for more steps...

Step 13.2.1

Multiply $5$ by $- 8$ .

$\frac{21}{- 40 - \sqrt{25 {(- 8)}^{2} - 8 - 1 \cdot 5}}$

Step 13.2.2

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

$\frac{21}{- 40 - \sqrt{25 \cdot 64 - 8 - 1 \cdot 5}}$

Step 13.2.3

Multiply $25$ by $64$ .

$\frac{21}{- 40 - \sqrt{1600 - 8 - 1 \cdot 5}}$

Step 13.2.4

Multiply $- 1$ by $5$ .

$\frac{21}{- 40 - \sqrt{1600 - 8 - 5}}$

Step 13.2.5

Subtract $8$ from $1600$ .

$\frac{21}{- 40 - \sqrt{1592 - 5}}$

Step 13.2.6

Subtract $5$ from $1592$ .

$\frac{21}{- 40 - \sqrt{1587}}$

Step 13.2.7

Rewrite $1587$ as $23^{2} \cdot 3$ .

Tap for more steps...

Step 13.2.7.1

Factor $529$ out of $1587$ .

$\frac{21}{- 40 - \sqrt{529 (3)}}$

Step 13.2.7.2

Rewrite $529$ as $23^{2}$ .

$\frac{21}{- 40 - \sqrt{23^{2} \cdot 3}}$

Step 13.2.8

Pull terms out from under the radical.

$\frac{21}{- 40 - (23 \sqrt{3})}$

Step 13.2.9

Multiply $23$ by $- 1$ .

$\frac{21}{- 40 - 23 \sqrt{3}}$

Step 13.3

Multiply $\frac{21}{- 40 - 23 \sqrt{3}}$ by $\frac{- 40 + 23 \sqrt{3}}{- 40 + 23 \sqrt{3}}$ .

$\frac{21}{- 40 - 23 \sqrt{3}} \cdot \frac{- 40 + 23 \sqrt{3}}{- 40 + 23 \sqrt{3}}$

Step 13.4

Multiply $\frac{21}{- 40 - 23 \sqrt{3}}$ by $\frac{- 40 + 23 \sqrt{3}}{- 40 + 23 \sqrt{3}}$ .

$\frac{21 (- 40 + 23 \sqrt{3})}{(- 40 - 23 \sqrt{3}) (- 40 + 23 \sqrt{3})}$

Step 13.5

Expand the denominator using the FOIL method.

$\frac{21 (- 40 + 23 \sqrt{3})}{1600 - 920 \sqrt{3} + 920 \sqrt{3} - 529 {\sqrt{3}}^{2}}$

Step 13.6

Simplify.

$\frac{21 (- 40 + 23 \sqrt{3})}{13}$

Step 13.7

Rewrite $- 40$ as $- 1 (40)$ .

$\frac{21 (- 1 (40) + 23 \sqrt{3})}{13}$

Step 13.8

Factor $- 1$ out of $23 \sqrt{3}$ .

$\frac{21 (- 1 (40) - (- 23 \sqrt{3}))}{13}$

Step 13.9

Factor $- 1$ out of $- 1 (40) - (- 23 \sqrt{3})$ .

$\frac{21 (- 1 (40 - 23 \sqrt{3}))}{13}$

Step 13.10

Move the negative in front of the fraction.

$- \frac{21 (40 - 23 \sqrt{3})}{13}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$- \frac{21 (40 - 23 \sqrt{3})}{13}$

Decimal Form:

$- 0.26303538 \dots$