Calculus Examples

$lim_{x \to 5} \frac{\sqrt{x + 5} - 3}{x - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sqrt{x + 5} - 3}{lim_{x \to 5} x - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sqrt{x + 5} - lim_{x \to 5} 3}{lim_{x \to 5} x - 4}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 5} x + 5} - lim_{x \to 5} 3}{lim_{x \to 5} x - 4}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + lim_{x \to 5} 5} - lim_{x \to 5} 3}{lim_{x \to 5} x - 4}$

Step 5

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

$\frac{\sqrt{lim_{x \to 5} x + 5} - lim_{x \to 5} 3}{lim_{x \to 5} x - 4}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 5} - 1 \cdot 3}{lim_{x \to 5} x - 4}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 5} - 1 \cdot 3}{lim_{x \to 5} x - lim_{x \to 5} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $5$ .

$\frac{\sqrt{lim_{x \to 5} x + 5} - 1 \cdot 3}{lim_{x \to 5} x - 1 \cdot 4}$

Step 9

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sqrt{5 + 5} - 1 \cdot 3}{lim_{x \to 5} x - 1 \cdot 4}$

Step 9.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Add $5$ and $5$ .

$\frac{\sqrt{10} - 1 \cdot 3}{5 - 1 \cdot 4}$

Step 10.1.2

Multiply $- 1$ by $3$ .

$\frac{\sqrt{10} - 3}{5 - 1 \cdot 4}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 1$ by $4$ .

$\frac{\sqrt{10} - 3}{5 - 4}$

Step 10.2.2

Subtract $4$ from $5$ .

$\frac{\sqrt{10} - 3}{1}$

Step 10.3

Divide $\sqrt{10} - 3$ by $1$ .

$\sqrt{10} - 3$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10} - 3$

Decimal Form:

$0.16227766 \dots$