Calculus Examples

$lim_{x \to 3.9} \frac{x - 4}{\sqrt{x} - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3.9$ .

$\frac{lim_{x \to 3.9} x - 4}{lim_{x \to 3.9} \sqrt{x} - 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.9$ .

$\frac{lim_{x \to 3.9} x - lim_{x \to 3.9} 4}{lim_{x \to 3.9} \sqrt{x} - 2}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $3.9$ .

$\frac{lim_{x \to 3.9} x - 1 \cdot 4}{lim_{x \to 3.9} \sqrt{x} - 2}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.9$ .

$\frac{lim_{x \to 3.9} x - 1 \cdot 4}{lim_{x \to 3.9} \sqrt{x} - lim_{x \to 3.9} 2}$

Step 5

Move the limit under the radical sign.

$\frac{lim_{x \to 3.9} x - 1 \cdot 4}{\sqrt{lim_{x \to 3.9} x} - lim_{x \to 3.9} 2}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $3.9$ .

$\frac{lim_{x \to 3.9} x - 1 \cdot 4}{\sqrt{lim_{x \to 3.9} x} - 1 \cdot 2}$

Step 7

Evaluate the limits by plugging in $3.9$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $3.9$ for $x$ .

$\frac{3.9 - 1 \cdot 4}{\sqrt{lim_{x \to 3.9} x} - 1 \cdot 2}$

Step 7.2

Evaluate the limit of $x$ by plugging in $3.9$ for $x$ .

$\frac{3.9 - 1 \cdot 4}{\sqrt{3.9} - 1 \cdot 2}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 1$ by $4$ .

$\frac{3.9 - 4}{\sqrt{3.9} - 1 \cdot 2}$

Step 8.1.2

Subtract $4$ from $3.9$ .

$\frac{- 0.1}{\sqrt{3.9} - 1 \cdot 2}$

Step 8.2

Multiply $- 1$ by $2$ .

$\frac{- 0.1}{\sqrt{3.9} - 2}$

Step 8.3

Move the negative in front of the fraction.

$- \frac{0.1}{\sqrt{3.9} - 2}$

Step 8.4

Multiply $\frac{0.1}{\sqrt{3.9} - 2}$ by $\frac{\sqrt{3.9} + 2}{\sqrt{3.9} + 2}$ .

$- (\frac{0.1}{\sqrt{3.9} - 2} \cdot \frac{\sqrt{3.9} + 2}{\sqrt{3.9} + 2})$

Step 8.5

Multiply $\frac{0.1}{\sqrt{3.9} - 2}$ by $\frac{\sqrt{3.9} + 2}{\sqrt{3.9} + 2}$ .

$- \frac{0.1 (\sqrt{3.9} + 2)}{(\sqrt{3.9} - 2) (\sqrt{3.9} + 2)}$

Step 8.6

Expand the denominator using the FOIL method.

$- \frac{0.1 (\sqrt{3.9} + 2)}{{\sqrt{3.9}}^{2} + \sqrt{3.9} \cdot 2 - 2 \sqrt{3.9} - 4}$

Step 8.7

Simplify.

$- \frac{0.1 (\sqrt{3.9} + 2)}{- 0.1}$

Step 8.8

Cancel the common factor of $0.1$ and $- 0.1$ .

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Step 8.8.1

Rewrite $0.1$ as $- 1 (- 0.1)$ .

$- \frac{- 1 (- 0.1) (\sqrt{3.9} + 2)}{- 0.1}$

Step 8.8.2

Cancel the common factor.

$- \frac{- 1 \cdot - 0.1 (\sqrt{3.9} + 2)}{- 0.1}$

Step 8.8.3

Divide $- (\sqrt{3.9} + 2)$ by $1$ .

$- - (\sqrt{3.9} + 2)$

Step 8.9

Apply the distributive property.

$- (- \sqrt{3.9} - 1 \cdot 2)$

Step 8.10

Multiply $- 1$ by $2$ .

$- (- \sqrt{3.9} - 2)$

Step 8.11

Apply the distributive property.

$- - \sqrt{3.9} - - 2$

Step 8.12

Multiply $- - \sqrt{3.9}$ .

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Step 8.12.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{3.9} - - 2$

Step 8.12.2

Multiply $\sqrt{3.9}$ by $1$ .

$\sqrt{3.9} - - 2$

Step 8.13

Multiply $- 1$ by $- 2$ .

$\sqrt{3.9} + 2$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3.9} + 2$

Decimal Form:

$3.97484176 \dots$