Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 1$ .

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

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $- 1$ to the power of $2$ .

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

Step 12.1.2

Add $1$ and $3$ .

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

Step 12.1.3

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

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

Step 12.1.4

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

$\frac{2 - 1 \cdot 2}{{(- 1)}^{2} + 1}$

Step 12.1.5

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{{(- 1)}^{2} + 1}$

Step 12.1.6

Subtract $2$ from $2$ .

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

Step 12.2

Simplify the denominator.

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Step 12.2.1

Raise $- 1$ to the power of $2$ .

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

Step 12.2.2

Add $1$ and $1$ .

$\frac{0}{2}$

Step 12.3

Divide $0$ by $2$ .

$0$