Calculus Examples

$lim_{x \to 2} \frac{\sqrt{2 x} - \sqrt{6 - x}}{4 + x^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{2 x} - \sqrt{6 - x}}{lim_{x \to 2} 4 + x^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{2 x} - lim_{x \to 2} \sqrt{6 - x}}{lim_{x \to 2} 4 + x^{2}}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} 2 x} - lim_{x \to 2} \sqrt{6 - x}}{lim_{x \to 2} 4 + x^{2}}$

Step 4

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

$\frac{\sqrt{2 lim_{x \to 2} x} - lim_{x \to 2} \sqrt{6 - x}}{lim_{x \to 2} 4 + x^{2}}$

Step 5

Move the limit under the radical sign.

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{lim_{x \to 2} 6 - x}}{lim_{x \to 2} 4 + x^{2}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{lim_{x \to 2} 6 - lim_{x \to 2} x}}{lim_{x \to 2} 4 + x^{2}}$

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{6 - lim_{x \to 2} x}}{lim_{x \to 2} 4 + x^{2}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{6 - lim_{x \to 2} x}}{lim_{x \to 2} 4 + lim_{x \to 2} x^{2}}$

Step 9

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

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{6 - lim_{x \to 2} x}}{4 + lim_{x \to 2} x^{2}}$

Step 10

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

$\frac{\sqrt{2 lim_{x \to 2} x} - \sqrt{6 - lim_{x \to 2} x}}{4 + {(lim_{x \to 2} x)}^{2}}$

Step 11

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2} - \sqrt{6 - lim_{x \to 2} x}}{4 + {(lim_{x \to 2} x)}^{2}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2} - \sqrt{6 - 2}}{4 + {(lim_{x \to 2} x)}^{2}}$

Step 11.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2} - \sqrt{6 - 2}}{4 + 2^{2}}$

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 $2$ .

$\frac{\sqrt{4} - \sqrt{6 - 2}}{4 + 2^{2}}$

Step 12.1.2

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

$\frac{\sqrt{2^{2}} - \sqrt{6 - 2}}{4 + 2^{2}}$

Step 12.1.3

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

$\frac{2 - \sqrt{6 - 2}}{4 + 2^{2}}$

Step 12.1.4

Subtract $2$ from $6$ .

$\frac{2 - \sqrt{4}}{4 + 2^{2}}$

Step 12.1.5

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

$\frac{2 - \sqrt{2^{2}}}{4 + 2^{2}}$

Step 12.1.6

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

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

Step 12.1.7

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{4 + 2^{2}}$

Step 12.1.8

Subtract $2$ from $2$ .

$\frac{0}{4 + 2^{2}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

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

$\frac{0}{4 + 4}$

Step 12.2.2

Add $4$ and $4$ .

$\frac{0}{8}$

Step 12.3

Divide $0$ by $8$ .

$0$