Calculus Examples

$lim_{x \to 3} \frac{\sqrt{x + 1}}{x - 4}$

Step 1

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

$\frac{lim_{x \to 3} \sqrt{x + 1}}{lim_{x \to 3} x - 4}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 3} x + 1}}{lim_{x \to 3} x - 4}$

Step 3

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

$\frac{\sqrt{lim_{x \to 3} x + lim_{x \to 3} 1}}{lim_{x \to 3} x - 4}$

Step 4

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

$\frac{\sqrt{lim_{x \to 3} x + 1}}{lim_{x \to 3} x - 4}$

Step 5

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

$\frac{\sqrt{lim_{x \to 3} x + 1}}{lim_{x \to 3} x - lim_{x \to 3} 4}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{lim_{x \to 3} x + 1}}{lim_{x \to 3} x - 1 \cdot 4}$

Step 7

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

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Step 7.1

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

$\frac{\sqrt{3 + 1}}{lim_{x \to 3} x - 1 \cdot 4}$

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add $3$ and $1$ .

$\frac{\sqrt{4}}{3 - 1 \cdot 4}$

Step 8.1.2

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

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

Step 8.1.3

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

$\frac{2}{3 - 1 \cdot 4}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $4$ .

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

Step 8.2.2

Subtract $4$ from $3$ .

$\frac{2}{- 1}$

Step 8.3

Divide $2$ by $- 1$ .

$- 2$