Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.4

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

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

Step 1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

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

Step 1.6

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$\frac{\sqrt{{(lim_{x \to 1} x)}^{2} + 3} - 1 \cdot 2}{x^{2} - 1}$

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Add $1$ and $3$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

$\frac{2 - 1 \cdot 2}{x^{2} - 1}$

Step 3.1.5

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{x^{2} - 1}$

Step 3.1.6

Subtract $2$ from $2$ .

$\frac{0}{x^{2} - 1}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{0}{x^{2} - 1^{2}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 3.3

Divide $0$ by $(x + 1) (x - 1)$ .

$0$