Calculus Examples

$lim_{x \to 2} \sqrt{\frac{x^{2} + 3 x + 4}{x^{3} + 1}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 2} \frac{x^{2} + 3 x + 4}{x^{3} + 1}}$

Step 2

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

$\sqrt{\frac{lim_{x \to 2} x^{2} + 3 x + 4}{lim_{x \to 2} x^{3} + 1}}$

Step 3

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

$\sqrt{\frac{lim_{x \to 2} x^{2} + lim_{x \to 2} 3 x + lim_{x \to 2} 4}{lim_{x \to 2} x^{3} + 1}}$

Step 4

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

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 3 x + lim_{x \to 2} 4}{lim_{x \to 2} x^{3} + 1}}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x + lim_{x \to 2} 4}{lim_{x \to 2} x^{3} + 1}}$

Step 6

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

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x + 4}{lim_{x \to 2} x^{3} + 1}}$

Step 7

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

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x + 4}{lim_{x \to 2} x^{3} + lim_{x \to 2} 1}}$

Step 8

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x + 4}{{(lim_{x \to 2} x)}^{3} + lim_{x \to 2} 1}}$

Step 9

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

$\sqrt{\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x + 4}{{(lim_{x \to 2} x)}^{3} + 1}}$

Step 10

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

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Step 10.1

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

$\sqrt{\frac{2^{2} + 3 lim_{x \to 2} x + 4}{{(lim_{x \to 2} x)}^{3} + 1}}$

Step 10.2

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

$\sqrt{\frac{2^{2} + 3 \cdot 2 + 4}{{(lim_{x \to 2} x)}^{3} + 1}}$

Step 10.3

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

$\sqrt{\frac{2^{2} + 3 \cdot 2 + 4}{2^{3} + 1}}$

Step 11

Simplify the answer.

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Step 11.1

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

$\sqrt{\frac{4 + 3 \cdot 2 + 4}{2^{3} + 1}}$

Step 11.2

Multiply $3$ by $2$ .

$\sqrt{\frac{4 + 6 + 4}{2^{3} + 1}}$

Step 11.3

Add $4$ and $6$ .

$\sqrt{\frac{10 + 4}{2^{3} + 1}}$

Step 11.4

Add $10$ and $4$ .

$\sqrt{\frac{14}{2^{3} + 1}}$

Step 11.5

Raise $2$ to the power of $3$ .

$\sqrt{\frac{14}{8 + 1}}$

Step 11.6

Add $8$ and $1$ .

$\sqrt{\frac{14}{9}}$

Step 11.7

Rewrite $\sqrt{\frac{14}{9}}$ as $\frac{\sqrt{14}}{\sqrt{9}}$ .

$\frac{\sqrt{14}}{\sqrt{9}}$

Step 11.8

Simplify the denominator.

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Step 11.8.1

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{14}}{\sqrt{3^{2}}}$

Step 11.8.2

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

$\frac{\sqrt{14}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{14}}{3}$

Decimal Form:

$1.24721912 \dots$