Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

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 $x$ .

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 11.1.1

Multiply $2$ by $2^{2}$ .

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Step 11.1.1.1

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

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

Step 11.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.1.2

Add $1$ and $2$ .

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

Step 11.2

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

$\sqrt{\frac{8 + 1}{3 \cdot 2 - 1 \cdot 2}}$

Step 11.3

Add $8$ and $1$ .

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

Step 11.4

Multiply $3$ by $2$ .

$\sqrt{\frac{9}{6 - 1 \cdot 2}}$

Step 11.5

Multiply $- 1$ by $2$ .

$\sqrt{\frac{9}{6 - 2}}$

Step 11.6

Subtract $2$ from $6$ .

$\sqrt{\frac{9}{4}}$

Step 11.7

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$\frac{\sqrt{9}}{\sqrt{4}}$

Step 11.8

Simplify the numerator.

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Step 11.8.1

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

$\frac{\sqrt{3^{2}}}{\sqrt{4}}$

Step 11.8.2

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

$\frac{3}{\sqrt{4}}$

Step 11.9

Simplify the denominator.

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Step 11.9.1

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

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

Step 11.9.2

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

$\frac{3}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$