Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Move the limit under the radical sign.

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

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 7

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

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

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $3$ .

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

Step 9

Move the limit under the radical sign.

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

Step 10

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 11

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

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

Step 12

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 12.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 12.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{9 + 3 - \sqrt{3^{3} - 1 \cdot 2}}{\sqrt{3 + 1}}$

Step 13.1.2

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

$\frac{9 + 3 - \sqrt{27 - 1 \cdot 2}}{\sqrt{3 + 1}}$

Step 13.1.3

Multiply $- 1$ by $2$ .

$\frac{9 + 3 - \sqrt{27 - 2}}{\sqrt{3 + 1}}$

Step 13.1.4

Subtract $2$ from $27$ .

$\frac{9 + 3 - \sqrt{25}}{\sqrt{3 + 1}}$

Step 13.1.5

Rewrite $25$ as $5^{2}$ .

$\frac{9 + 3 - \sqrt{5^{2}}}{\sqrt{3 + 1}}$

Step 13.1.6

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

$\frac{9 + 3 - 1 \cdot 5}{\sqrt{3 + 1}}$

Step 13.1.7

Multiply $- 1$ by $5$ .

$\frac{9 + 3 - 5}{\sqrt{3 + 1}}$

Step 13.1.8

Add $9$ and $3$ .

$\frac{12 - 5}{\sqrt{3 + 1}}$

Step 13.1.9

Subtract $5$ from $12$ .

$\frac{7}{\sqrt{3 + 1}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Add $3$ and $1$ .

$\frac{7}{\sqrt{4}}$

Step 13.2.2

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

$\frac{7}{\sqrt{2^{2}}}$

Step 13.2.3

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

$\frac{7}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{2}$

Decimal Form:

$3.5$