Calculus Examples

$lim_{x \to 3} \frac{\sqrt{x^{2} + 7} + \sqrt{3 x - 5}}{x + 2}$

Step 1

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

$\frac{lim_{x \to 3} \sqrt{x^{2} + 7} + \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 2

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

$\frac{lim_{x \to 3} \sqrt{x^{2} + 7} + lim_{x \to 3} \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 3} x^{2} + 7} + lim_{x \to 3} \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 4

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

$\frac{\sqrt{lim_{x \to 3} x^{2} + lim_{x \to 3} 7} + lim_{x \to 3} \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 5

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

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 7} + lim_{x \to 3} \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + lim_{x \to 3} \sqrt{3 x - 5}}{lim_{x \to 3} x + 2}$

Step 7

Move the limit under the radical sign.

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{lim_{x \to 3} 3 x - 5}}{lim_{x \to 3} x + 2}$

Step 8

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

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{lim_{x \to 3} 3 x - lim_{x \to 3} 5}}{lim_{x \to 3} x + 2}$

Step 9

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

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{3 lim_{x \to 3} x - lim_{x \to 3} 5}}{lim_{x \to 3} x + 2}$

Step 10

Evaluate the limit of $5$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{3 lim_{x \to 3} x - 1 \cdot 5}}{lim_{x \to 3} x + 2}$

Step 11

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

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{3 lim_{x \to 3} x - 1 \cdot 5}}{lim_{x \to 3} x + lim_{x \to 3} 2}$

Step 12

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

$\frac{\sqrt{{(lim_{x \to 3} x)}^{2} + 7} + \sqrt{3 lim_{x \to 3} x - 1 \cdot 5}}{lim_{x \to 3} x + 2}$

Step 13

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

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Step 13.1

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

$\frac{\sqrt{3^{2} + 7} + \sqrt{3 lim_{x \to 3} x - 1 \cdot 5}}{lim_{x \to 3} x + 2}$

Step 13.2

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

$\frac{\sqrt{3^{2} + 7} + \sqrt{3 \cdot 3 - 1 \cdot 5}}{lim_{x \to 3} x + 2}$

Step 13.3

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

$\frac{\sqrt{3^{2} + 7} + \sqrt{3 \cdot 3 - 1 \cdot 5}}{3 + 2}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

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

$\frac{\sqrt{9 + 7} + \sqrt{3 \cdot 3 - 1 \cdot 5}}{3 + 2}$

Step 14.1.2

Add $9$ and $7$ .

$\frac{\sqrt{16} + \sqrt{3 \cdot 3 - 1 \cdot 5}}{3 + 2}$

Step 14.1.3

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}} + \sqrt{3 \cdot 3 - 1 \cdot 5}}{3 + 2}$

Step 14.1.4

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

$\frac{4 + \sqrt{3 \cdot 3 - 1 \cdot 5}}{3 + 2}$

Step 14.1.5

Multiply $3$ by $3$ .

$\frac{4 + \sqrt{9 - 1 \cdot 5}}{3 + 2}$

Step 14.1.6

Multiply $- 1$ by $5$ .

$\frac{4 + \sqrt{9 - 5}}{3 + 2}$

Step 14.1.7

Subtract $5$ from $9$ .

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

Step 14.1.8

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

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

Step 14.1.9

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

$\frac{4 + 2}{3 + 2}$

Step 14.1.10

Add $4$ and $2$ .

$\frac{6}{3 + 2}$

Step 14.2

Add $3$ and $2$ .

$\frac{6}{5}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{5}$

Decimal Form:

$1.2$