Calculus Examples

$lim_{x \to 3.999} \frac{4 - \sqrt{x + 12}}{x - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3.999$ .

$\frac{lim_{x \to 3.999} 4 - \sqrt{x + 12}}{lim_{x \to 3.999} x - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.999$ .

$\frac{lim_{x \to 3.999} 4 - lim_{x \to 3.999} \sqrt{x + 12}}{lim_{x \to 3.999} x - 4}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $3.999$ .

$\frac{4 - lim_{x \to 3.999} \sqrt{x + 12}}{lim_{x \to 3.999} x - 4}$

Step 4

Move the limit under the radical sign.

$\frac{4 - \sqrt{lim_{x \to 3.999} x + 12}}{lim_{x \to 3.999} x - 4}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.999$ .

$\frac{4 - \sqrt{lim_{x \to 3.999} x + lim_{x \to 3.999} 12}}{lim_{x \to 3.999} x - 4}$

Step 6

Evaluate the limit of $12$ which is constant as $x$ approaches $3.999$ .

$\frac{4 - \sqrt{lim_{x \to 3.999} x + 12}}{lim_{x \to 3.999} x - 4}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.999$ .

$\frac{4 - \sqrt{lim_{x \to 3.999} x + 12}}{lim_{x \to 3.999} x - lim_{x \to 3.999} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $3.999$ .

$\frac{4 - \sqrt{lim_{x \to 3.999} x + 12}}{lim_{x \to 3.999} x - 1 \cdot 4}$

Step 9

Evaluate the limits by plugging in $3.999$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $3.999$ for $x$ .

$\frac{4 - \sqrt{3.999 + 12}}{lim_{x \to 3.999} x - 1 \cdot 4}$

Step 9.2

Evaluate the limit of $x$ by plugging in $3.999$ for $x$ .

$\frac{4 - \sqrt{3.999 + 12}}{3.999 - 1 \cdot 4}$

Step 10

Simplify the answer.

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Step 10.1

Add $3.999$ and $12$ .

$\frac{4 - \sqrt{15.999}}{3.999 - 1 \cdot 4}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 1$ by $4$ .

$\frac{4 - \sqrt{15.999}}{3.999 - 4}$

Step 10.2.2

Subtract $4$ from $3.999$ .

$\frac{4 - \sqrt{15.999}}{- 0.001}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{4 - \sqrt{15.999}}{0.001}$

Step 10.4

Evaluate the root.

$- \frac{4 - 1 \cdot 3.99987499}{0.001}$

Step 10.5

Multiply $- 1$ by $3.99987499$ .

$- \frac{4 - 3.99987499}{0.001}$

Step 10.6

Subtract $3.99987499$ from $4$ .

$- \frac{0.000125}{0.001}$

Step 10.7

Divide $0.000125$ by $0.001$ .

$- 1 \cdot 0.12500195$

Step 10.8

Multiply $- 1$ by $0.12500195$ .

$- 0.12500195$