Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $2$ by $2$ .

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

Step 10.1.2

Add $4$ and $5$ .

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

Step 10.1.3

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

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

Step 10.1.4

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

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

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 3$ by $2$ .

$\frac{3}{1 - 6}$

Step 10.2.2

Subtract $6$ from $1$ .

$\frac{3}{- 5}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{3}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{5}$

Decimal Form:

$- 0.6$