Calculus Examples

$lim_{x \to 4} \frac{\sqrt{2 x + 1} - 3}{\sqrt{x - 2 \sqrt{2}}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} \sqrt{2 x + 1} - 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} \sqrt{2 x + 1} - lim_{x \to 4} 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 4} 2 x + 1} - lim_{x \to 4} 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{\sqrt{lim_{x \to 4} 2 x + lim_{x \to 4} 1} - lim_{x \to 4} 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 5

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

$\frac{\sqrt{2 lim_{x \to 4} x + lim_{x \to 4} 1} - lim_{x \to 4} 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1} - lim_{x \to 4} 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1} - 1 \cdot 3}{lim_{x \to 4} \sqrt{x - 2 \sqrt{2}}}$

Step 8

Move the limit under the radical sign.

$\frac{\sqrt{2 lim_{x \to 4} x + 1} - 1 \cdot 3}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1} - 1 \cdot 3}{\sqrt{lim_{x \to 4} x - lim_{x \to 4} 2 \sqrt{2}}}$

Step 10

Evaluate the limit of $2 \sqrt{2}$ which is constant as $x$ approaches $4$ .

$\frac{\sqrt{2 lim_{x \to 4} x + 1} - 1 \cdot 3}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 11

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{\sqrt{2 \cdot 4 + 1} - 1 \cdot 3}{\sqrt{lim_{x \to 4} x - 2 \sqrt{2}}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{\sqrt{2 \cdot 4 + 1} - 1 \cdot 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $2$ by $4$ .

$\frac{\sqrt{8 + 1} - 1 \cdot 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.1.2

Add $8$ and $1$ .

$\frac{\sqrt{9} - 1 \cdot 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.1.3

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

$\frac{\sqrt{3^{2}} - 1 \cdot 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.1.4

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

$\frac{3 - 1 \cdot 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.1.5

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.1.6

Subtract $3$ from $3$ .

$\frac{0}{\sqrt{4 - 2 \sqrt{2}}}$

Step 12.2

Divide $0$ by $\sqrt{4 - 2 \sqrt{2}}$ .

$0$