Calculus Examples

$lim_{x \to 4} \frac{8 \sqrt{x} - x^{2}}{8 - 4 \sqrt{x} c}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 8 \sqrt{x} - x^{2}}{lim_{x \to 4} 8 - 4 \sqrt{x} c}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 8 \sqrt{x} - lim_{x \to 4} x^{2}}{lim_{x \to 4} 8 - 4 \sqrt{x} c}$

Step 3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 lim_{x \to 4} \sqrt{x} - lim_{x \to 4} x^{2}}{lim_{x \to 4} 8 - 4 \sqrt{x} c}$

Step 4

Move the limit under the radical sign.

$\frac{8 \sqrt{lim_{x \to 4} x} - lim_{x \to 4} x^{2}}{lim_{x \to 4} 8 - 4 \sqrt{x} c}$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{8 \sqrt{lim_{x \to 4} x} - {(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} 8 - 4 \sqrt{x} c}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{8 \sqrt{lim_{x \to 4} x} - {(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} 8 - lim_{x \to 4} 4 \sqrt{x} c}$

Step 7

Evaluate the limit of $8$ which is constant as $x$ approaches $4$ .

$\frac{8 \sqrt{lim_{x \to 4} x} - {(lim_{x \to 4} x)}^{2}}{8 - lim_{x \to 4} 4 \sqrt{x} c}$

Step 8

Move the term $4 c$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 \sqrt{lim_{x \to 4} x} - {(lim_{x \to 4} x)}^{2}}{8 - 1 \cdot 4 c lim_{x \to 4} \sqrt{x}}$

Step 9

Move the limit under the radical sign.

$\frac{8 \sqrt{lim_{x \to 4} x} - {(lim_{x \to 4} x)}^{2}}{8 - 1 \cdot 4 c \sqrt{lim_{x \to 4} x}}$

Step 10

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{8 \sqrt{4} - {(lim_{x \to 4} x)}^{2}}{8 - 1 \cdot 4 c \sqrt{lim_{x \to 4} x}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{8 \sqrt{4} - 4^{2}}{8 - 1 \cdot 4 c \sqrt{lim_{x \to 4} x}}$

Step 10.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{8 \sqrt{4} - 4^{2}}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{8 \sqrt{2^{2}} - 4^{2}}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.1.2

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

$\frac{8 \cdot 2 - 4^{2}}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.1.3

Multiply $8$ by $2$ .

$\frac{16 - 4^{2}}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.1.4

Raise $4$ to the power of $2$ .

$\frac{16 - 1 \cdot 16}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.1.5

Multiply $- 1$ by $16$ .

$\frac{16 - 16}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.1.6

Subtract $16$ from $16$ .

$\frac{0}{8 - 1 \cdot 4 c \sqrt{4}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 1$ by $4$ .

$\frac{0}{8 - 4 c \sqrt{4}}$

Step 11.2.2

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

$\frac{0}{8 - 4 c \sqrt{2^{2}}}$

Step 11.2.3

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

$\frac{0}{8 - 4 c \cdot 2}$

Step 11.2.4

Multiply $2$ by $- 4$ .

$\frac{0}{8 - 8 c}$

Step 11.2.5

Factor $8$ out of $8 - 8 c$ .

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Step 11.2.5.1

Factor $8$ out of $8$ .

$\frac{0}{8 (1) - 8 c}$

Step 11.2.5.2

Factor $8$ out of $- 8 c$ .

$\frac{0}{8 (1) + 8 (- c)}$

Step 11.2.5.3

Factor $8$ out of $8 (1) + 8 (- c)$ .

$\frac{0}{8 (1 - c)}$

Step 11.3

Cancel the common factor of $0$ and $8$ .

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Step 11.3.1

Factor $8$ out of $0$ .

$\frac{8 (0)}{8 (1 - c)}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Cancel the common factor.

$\frac{8 \cdot 0}{8 (1 - c)}$

Step 11.3.2.2

Rewrite the expression.

$\frac{0}{1 - c}$

Step 11.4

Divide $0$ by $1 - c$ .

$0$