Calculus Examples

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

Step 1

Evaluate the limit of $\frac{4 - y}{2 - \sqrt{y}}$ which is constant as $x$ approaches $4$ .

$\frac{4 - y}{2 - \sqrt{y}}$

Step 2

Simplify the answer.

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Step 2.1

Multiply $\frac{4 - y}{2 - \sqrt{y}}$ by $\frac{2 + \sqrt{y}}{2 + \sqrt{y}}$ .

$\frac{4 - y}{2 - \sqrt{y}} \cdot \frac{2 + \sqrt{y}}{2 + \sqrt{y}}$

Step 2.2

Multiply $\frac{4 - y}{2 - \sqrt{y}}$ by $\frac{2 + \sqrt{y}}{2 + \sqrt{y}}$ .

$\frac{(4 - y) (2 + \sqrt{y})}{(2 - \sqrt{y}) (2 + \sqrt{y})}$

Step 2.3

Expand the denominator using the FOIL method.

$\frac{(4 - y) (2 + \sqrt{y})}{4 + 2 \sqrt{y} - 2 \sqrt{y} - {\sqrt{y}}^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{(4 - y) (2 + \sqrt{y})}{4 - {(y^{\frac{1}{2}})}^{2}}$

Step 2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y^{\frac{1}{2} \cdot 2}}$

Step 2.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y^{\frac{2}{2}}}$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y^{\frac{2}{2}}}$

Step 2.4.4.2

Rewrite the expression.

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y^{1}}$

Step 2.4.5

Simplify.

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y}$

Step 2.5

Cancel the common factor of $4 - y$ .

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Step 2.5.1

Cancel the common factor.

$\frac{(4 - y) (2 + \sqrt{y})}{4 - y}$

Step 2.5.2

Divide $2 + \sqrt{y}$ by $1$ .

$2 + \sqrt{y}$