Calculus Examples

$lim_{x \to 1} \sqrt{\frac{8 x + 1}{x + 3}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 1} \frac{8 x + 1}{x + 3}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\sqrt{\frac{lim_{x \to 1} 8 x + 1}{lim_{x \to 1} x + 3}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\sqrt{\frac{lim_{x \to 1} 8 x + lim_{x \to 1} 1}{lim_{x \to 1} x + 3}}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{8 lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x + 3}}$

Step 5

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

$\sqrt{\frac{8 lim_{x \to 1} x + 1}{lim_{x \to 1} x + 3}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\sqrt{\frac{8 lim_{x \to 1} x + 1}{lim_{x \to 1} x + lim_{x \to 1} 3}}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\sqrt{\frac{8 lim_{x \to 1} x + 1}{lim_{x \to 1} x + 3}}$

Step 8

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\sqrt{\frac{8 \cdot 1 + 1}{lim_{x \to 1} x + 3}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 9

Simplify the answer.

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Step 9.1

Multiply $8$ by $1$ .

$\sqrt{\frac{8 + 1}{1 + 3}}$

Step 9.2

Add $8$ and $1$ .

$\sqrt{\frac{9}{1 + 3}}$

Step 9.3

Add $1$ and $3$ .

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

Step 9.4

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

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

Step 9.5

Simplify the numerator.

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Step 9.5.1

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

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

Step 9.5.2

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

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

Step 9.6

Simplify the denominator.

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Step 9.6.1

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

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

Step 9.6.2

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

$\frac{3}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$