Calculus Examples

$lim_{x \to 5} - \frac{6 x}{\sqrt{8 x - 15}}$

Step 1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 5} \frac{6 x}{\sqrt{8 x - 15}}$

Step 2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$- 6 lim_{x \to 5} \frac{x}{\sqrt{8 x - 15}}$

Step 3

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

$- 6 \frac{lim_{x \to 5} x}{lim_{x \to 5} \sqrt{8 x - 15}}$

Step 4

Move the limit under the radical sign.

$- 6 \frac{lim_{x \to 5} x}{\sqrt{lim_{x \to 5} 8 x - 15}}$

Step 5

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

$- 6 \frac{lim_{x \to 5} x}{\sqrt{lim_{x \to 5} 8 x - lim_{x \to 5} 15}}$

Step 6

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$- 6 \frac{lim_{x \to 5} x}{\sqrt{8 lim_{x \to 5} x - lim_{x \to 5} 15}}$

Step 7

Evaluate the limit of $15$ which is constant as $x$ approaches $5$ .

$- 6 \frac{lim_{x \to 5} x}{\sqrt{8 lim_{x \to 5} x - 1 \cdot 15}}$

Step 8

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

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Step 8.1

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

$- 6 \frac{5}{\sqrt{8 lim_{x \to 5} x - 1 \cdot 15}}$

Step 8.2

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

$- 6 \frac{5}{\sqrt{8 \cdot 5 - 1 \cdot 15}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Multiply $8$ by $5$ .

$- 6 \frac{5}{\sqrt{40 - 1 \cdot 15}}$

Step 9.1.2

Multiply $- 1$ by $15$ .

$- 6 \frac{5}{\sqrt{40 - 15}}$

Step 9.1.3

Subtract $15$ from $40$ .

$- 6 \frac{5}{\sqrt{25}}$

Step 9.1.4

Rewrite $25$ as $5^{2}$ .

$- 6 \frac{5}{\sqrt{5^{2}}}$

Step 9.1.5

Pull terms out from under the radical, assuming positive real numbers.

$- 6 (\frac{5}{5})$

Step 9.2

Cancel the common factor of $5$ .

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Step 9.2.1

Cancel the common factor.

$- 6 (\frac{5}{5})$

Step 9.2.2

Rewrite the expression.

$- 6 \cdot 1$

Step 9.3

Multiply $- 6$ by $1$ .

$- 6$