Calculus Examples

$lim_{x \to 2} \sqrt{\frac{2 x^{2} - 5 x + 7}{x^{2} + 5}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 2} \frac{2 x^{2} - 5 x + 7}{x^{2} + 5}}$

Step 2

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

$\sqrt{\frac{lim_{x \to 2} 2 x^{2} - 5 x + 7}{lim_{x \to 2} x^{2} + 5}}$

Step 3

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

$\sqrt{\frac{lim_{x \to 2} 2 x^{2} - lim_{x \to 2} 5 x + lim_{x \to 2} 7}{lim_{x \to 2} x^{2} + 5}}$

Step 4

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

$\sqrt{\frac{2 lim_{x \to 2} x^{2} - lim_{x \to 2} 5 x + lim_{x \to 2} 7}{lim_{x \to 2} x^{2} + 5}}$

Step 5

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

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 5 x + lim_{x \to 2} 7}{lim_{x \to 2} x^{2} + 5}}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - 5 lim_{x \to 2} x + lim_{x \to 2} 7}{lim_{x \to 2} x^{2} + 5}}$

Step 7

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - 5 lim_{x \to 2} x + 7}{lim_{x \to 2} x^{2} + 5}}$

Step 8

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

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - 5 lim_{x \to 2} x + 7}{lim_{x \to 2} x^{2} + lim_{x \to 2} 5}}$

Step 9

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

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - 5 lim_{x \to 2} x + 7}{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 5}}$

Step 10

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\sqrt{\frac{2 {(lim_{x \to 2} x)}^{2} - 5 lim_{x \to 2} x + 7}{{(lim_{x \to 2} x)}^{2} + 5}}$

Step 11

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

Tap for more steps...

Step 11.1

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

$\sqrt{\frac{2 \cdot 2^{2} - 5 lim_{x \to 2} x + 7}{{(lim_{x \to 2} x)}^{2} + 5}}$

Step 11.2

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

$\sqrt{\frac{2 \cdot 2^{2} - 5 \cdot 2 + 7}{{(lim_{x \to 2} x)}^{2} + 5}}$

Step 11.3

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

$\sqrt{\frac{2 \cdot 2^{2} - 5 \cdot 2 + 7}{2^{2} + 5}}$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

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

Tap for more steps...

Step 12.1.1.1

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

$\sqrt{\frac{2^{1} \cdot 2^{2} - 5 \cdot 2 + 7}{2^{2} + 5}}$

Step 12.1.1.2

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

$\sqrt{\frac{2^{1 + 2} - 5 \cdot 2 + 7}{2^{2} + 5}}$

Step 12.1.2

Add $1$ and $2$ .

$\sqrt{\frac{2^{3} - 5 \cdot 2 + 7}{2^{2} + 5}}$

Step 12.2

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

$\sqrt{\frac{8 - 5 \cdot 2 + 7}{2^{2} + 5}}$

Step 12.3

Multiply $- 5$ by $2$ .

$\sqrt{\frac{8 - 10 + 7}{2^{2} + 5}}$

Step 12.4

Subtract $10$ from $8$ .

$\sqrt{\frac{- 2 + 7}{2^{2} + 5}}$

Step 12.5

Add $- 2$ and $7$ .

$\sqrt{\frac{5}{2^{2} + 5}}$

Step 12.6

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

$\sqrt{\frac{5}{4 + 5}}$

Step 12.7

Add $4$ and $5$ .

$\sqrt{\frac{5}{9}}$

Step 12.8

Rewrite $\sqrt{\frac{5}{9}}$ as $\frac{\sqrt{5}}{\sqrt{9}}$ .

$\frac{\sqrt{5}}{\sqrt{9}}$

Step 12.9

Simplify the denominator.

Tap for more steps...

Step 12.9.1

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

$\frac{\sqrt{5}}{\sqrt{3^{2}}}$

Step 12.9.2

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

$\frac{\sqrt{5}}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{3}$

Decimal Form:

$0.74535599 \dots$