Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

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 $x$ .

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

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Multiply $3$ by $4$ .

$\sqrt{\frac{12 + 4}{3 \cdot 2 - 1 \cdot 2}}$

Step 11.3

Add $12$ and $4$ .

$\sqrt{\frac{16}{3 \cdot 2 - 1 \cdot 2}}$

Step 11.4

Multiply $3$ by $2$ .

$\sqrt{\frac{16}{6 - 1 \cdot 2}}$

Step 11.5

Multiply $- 1$ by $2$ .

$\sqrt{\frac{16}{6 - 2}}$

Step 11.6

Subtract $2$ from $6$ .

$\sqrt{\frac{16}{4}}$

Step 11.7

Divide $16$ by $4$ .

$\sqrt{4}$

Step 11.8

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

$\sqrt{2^{2}}$

Step 11.9

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

$2$