Precalculus Examples

$\frac{lim_{x \to 1} \sqrt{8 x + 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 1} 8 x + 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 1.2

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

$\frac{\sqrt{lim_{x \to 1} 8 x + lim_{x \to 1} 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 1.3

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

$\frac{\sqrt{8 lim_{x \to 1} x + lim_{x \to 1} 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 1.4

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

$\frac{\sqrt{8 lim_{x \to 1} x + 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 2

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

$\frac{\sqrt{8 \cdot 1 + 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $8$ by $1$ .

$\frac{\sqrt{8 + 1}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 3.2

Add $8$ and $1$ .

$\frac{\sqrt{9}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 3.3

Simplify each term.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

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

$\frac{\sqrt{3^{2}}}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 3.3.1.2

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

$\frac{3}{2 (x - 1)} + \frac{3 x}{1 - x^{2}}$

Step 3.3.2

Simplify the denominator.

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Step 3.3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{3}{2 (x - 1)} + \frac{3 x}{1^{2} - x^{2}}$

Step 3.3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{3}{2 (x - 1)} + \frac{3 x}{(1 + x) (1 - x)}$