Precalculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the limit under the radical sign.

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $2$ .

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

Subtract $1$ from $- 2$ .

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

Step 3.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.2.2

Write the factored form using these integers.

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