Calculus Examples

$lim_{x \to 5} \frac{5 - \sqrt{6 - x}}{x}$

Step 1

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

$\frac{lim_{x \to 5} 5 - \sqrt{6 - x}}{lim_{x \to 5} x}$

Step 2

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

$\frac{lim_{x \to 5} 5 - lim_{x \to 5} \sqrt{6 - x}}{lim_{x \to 5} x}$

Step 3

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

$\frac{5 - lim_{x \to 5} \sqrt{6 - x}}{lim_{x \to 5} x}$

Step 4

Move the limit under the radical sign.

$\frac{5 - \sqrt{lim_{x \to 5} 6 - x}}{lim_{x \to 5} x}$

Step 5

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

$\frac{5 - \sqrt{lim_{x \to 5} 6 - lim_{x \to 5} x}}{lim_{x \to 5} x}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $5$ .

$\frac{5 - \sqrt{6 - lim_{x \to 5} x}}{lim_{x \to 5} x}$

Step 7

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

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Step 7.1

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

$\frac{5 - \sqrt{6 - 5}}{lim_{x \to 5} x}$

Step 7.2

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

$\frac{5 - \sqrt{6 - 5}}{5}$

Step 8

Simplify the numerator.

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Step 8.1

Subtract $5$ from $6$ .

$\frac{5 - \sqrt{1}}{5}$

Step 8.2

Any root of $1$ is $1$ .

$\frac{5 - 1 \cdot 1}{5}$

Step 8.3

Multiply $- 1$ by $1$ .

$\frac{5 - 1}{5}$

Step 8.4

Subtract $1$ from $5$ .

$\frac{4}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$