Calculus Examples

$\frac{lim_{x \to 2} x^{2} + 2 x - 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 1

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

$\frac{lim_{x \to 2} x^{2} + lim_{x \to 2} 2 x - lim_{x \to 2} 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 2

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

$\frac{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 2 x - lim_{x \to 2} 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 3

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

$\frac{{(lim_{x \to 2} x)}^{2} + 2 lim_{x \to 2} x - lim_{x \to 2} 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 4

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} + 2 lim_{x \to 2} x - 1 \cdot 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 5

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

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Step 5.1

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

$\frac{2^{2} + 2 lim_{x \to 2} x - 1 \cdot 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 5.2

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

$\frac{2^{2} + 2 \cdot 2 - 1 \cdot 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{4 + 2 \cdot 2 - 1 \cdot 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6.1.2

Multiply $2$ by $2$ .

$\frac{4 + 4 - 1 \cdot 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6.1.3

Multiply $- 1$ by $8$ .

$\frac{4 + 4 - 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6.1.4

Add $4$ and $4$ .

$\frac{8 - 8}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6.1.5

Subtract $8$ from $8$ .

$\frac{0}{\sqrt{x^{2}} + 5 - (x + 1)}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

$\frac{0}{x + 5 - (x + 1)}$

Step 6.2.2

Apply the distributive property.

$\frac{0}{x + 5 - x - 1 \cdot 1}$

Step 6.2.3

Multiply $- 1$ by $1$ .

$\frac{0}{x + 5 - x - 1}$

Step 6.2.4

Subtract $x$ from $x$ .

$\frac{0}{0 + 5 - 1}$

Step 6.2.5

Add $0$ and $5$ .

$\frac{0}{5 - 1}$

Step 6.2.6

Subtract $1$ from $5$ .

$\frac{0}{4}$

Step 6.3

Divide $0$ by $4$ .

$0$