Calculus Examples

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

Step 1

Cancel the common factor of $2 \sqrt{x} - 2$ and $2 x - 4$ .

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Step 1.1

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

$lim_{x \to 4} \frac{2 (\sqrt{x}) - 2}{2 x - 4}$

Step 1.2

Factor $2$ out of $- 2$ .

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

Step 1.3

Factor $2$ out of $2 (\sqrt{x}) + 2 (- 1)$ .

$lim_{x \to 4} \frac{2 (\sqrt{x} - 1)}{2 x - 4}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to 4} \frac{2 (\sqrt{x} - 1)}{2 (x) - 4}$

Step 1.4.2

Factor $2$ out of $- 4$ .

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

Step 1.4.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$lim_{x \to 4} \frac{2 (\sqrt{x} - 1)}{2 (x - 2)}$

Step 1.4.4

Cancel the common factor.

$lim_{x \to 4} \frac{2 (\sqrt{x} - 1)}{2 (x - 2)}$

Step 1.4.5

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

Multiply $- 1$ by $1$ .

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

Step 9.1.4

Subtract $1$ from $2$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $2$ .

$\frac{1}{4 - 2}$

Step 9.2.2

Subtract $2$ from $4$ .

$\frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$