Calculus Examples

$\frac{lim_{x \to 16} 4 - \sqrt{x}}{16 x - x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $16$ .

$\frac{lim_{x \to 16} 4 - lim_{x \to 16} \sqrt{x}}{16 x - x^{2}}$

Step 1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $16$ .

$\frac{4 - lim_{x \to 16} \sqrt{x}}{16 x - x^{2}}$

Step 1.3

Move the limit under the radical sign.

$\frac{4 - \sqrt{lim_{x \to 16} x}}{16 x - x^{2}}$

Step 2

Evaluate the limit of $x$ by plugging in $16$ for $x$ .

$\frac{4 - \sqrt{16}}{16 x - x^{2}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{4 - \sqrt{4^{2}}}{16 x - x^{2}}$

Step 3.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 - 1 \cdot 4}{16 x - x^{2}}$

Step 3.1.3

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{16 x - x^{2}}$

Step 3.1.4

Subtract $4$ from $4$ .

$\frac{0}{16 x - x^{2}}$

Step 3.2

Factor $x$ out of $16 x - x^{2}$ .

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Step 3.2.1

Factor $x$ out of $16 x$ .

$\frac{0}{x \cdot 16 - x^{2}}$

Step 3.2.2

Factor $x$ out of $- x^{2}$ .

$\frac{0}{x \cdot 16 + x (- x)}$

Step 3.2.3

Factor $x$ out of $x \cdot 16 + x (- x)$ .

$\frac{0}{x (16 - x)}$

Step 3.3

Divide $0$ by $x (16 - x)$ .

$0$