Calculus Examples

$lim_{x \to 5} \frac{3 - \sqrt{2 x - 1}}{3 (\sqrt{5 x - 5})}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 5} \frac{3 - \sqrt{2 x - 1}}{\sqrt{5 x - 5}}$

Step 2

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

$\frac{1}{3} \cdot \frac{lim_{x \to 5} 3 - \sqrt{2 x - 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 3

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

$\frac{1}{3} \cdot \frac{lim_{x \to 5} 3 - lim_{x \to 5} \sqrt{2 x - 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $5$ .

$\frac{1}{3} \cdot \frac{3 - lim_{x \to 5} \sqrt{2 x - 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 5

Move the limit under the radical sign.

$\frac{1}{3} \cdot \frac{3 - \sqrt{lim_{x \to 5} 2 x - 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 6

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{lim_{x \to 5} 2 x - lim_{x \to 5} 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 7

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - lim_{x \to 5} 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - 1 \cdot 1}}{lim_{x \to 5} \sqrt{5 x - 5}}$

Step 9

Move the limit under the radical sign.

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - 1 \cdot 1}}{\sqrt{lim_{x \to 5} 5 x - 5}}$

Step 10

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - 1 \cdot 1}}{\sqrt{lim_{x \to 5} 5 x - lim_{x \to 5} 5}}$

Step 11

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - 1 \cdot 1}}{\sqrt{5 lim_{x \to 5} x - lim_{x \to 5} 5}}$

Step 12

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 lim_{x \to 5} x - 1 \cdot 1}}{\sqrt{5 lim_{x \to 5} x - 1 \cdot 5}}$

Step 13

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

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Step 13.1

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 \cdot 5 - 1 \cdot 1}}{\sqrt{5 lim_{x \to 5} x - 1 \cdot 5}}$

Step 13.2

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

$\frac{1}{3} \cdot \frac{3 - \sqrt{2 \cdot 5 - 1 \cdot 1}}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $2$ by $5$ .

$\frac{1}{3} \cdot \frac{3 - \sqrt{10 - 1 \cdot 1}}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{3 - \sqrt{10 - 1}}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.3

Subtract $1$ from $10$ .

$\frac{1}{3} \cdot \frac{3 - \sqrt{9}}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.4

Rewrite $9$ as $3^{2}$ .

$\frac{1}{3} \cdot \frac{3 - \sqrt{3^{2}}}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.5

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

$\frac{1}{3} \cdot \frac{3 - 1 \cdot 3}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.6

Multiply $- 1$ by $3$ .

$\frac{1}{3} \cdot \frac{3 - 3}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.1.7

Subtract $3$ from $3$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{5 \cdot 5 - 1 \cdot 5}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Multiply $5$ by $5$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{25 - 1 \cdot 5}}$

Step 14.2.2

Multiply $- 1$ by $5$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{25 - 5}}$

Step 14.2.3

Subtract $5$ from $25$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{20}}$

Step 14.2.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 14.2.4.1

Factor $4$ out of $20$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{4 (5)}}$

Step 14.2.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{1}{3} \cdot \frac{0}{\sqrt{2^{2} \cdot 5}}$

Step 14.2.5

Pull terms out from under the radical.

$\frac{1}{3} \cdot \frac{0}{2 \sqrt{5}}$

Step 14.3

Cancel the common factor of $0$ and $2$ .

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Step 14.3.1

Factor $2$ out of $0$ .

$\frac{1}{3} \cdot \frac{2 (0)}{2 \sqrt{5}}$

Step 14.3.2

Cancel the common factors.

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Step 14.3.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{1}{3} \cdot \frac{2 (0)}{2 (\sqrt{5})}$

Step 14.3.2.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{2 \cdot 0}{2 \sqrt{5}}$

Step 14.3.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{0}{\sqrt{5}}$

Step 14.4

Divide $0$ by $\sqrt{5}$ .

$\frac{1}{3} \cdot 0$

Step 14.5

Multiply $\frac{1}{3}$ by $0$ .

$0$