Calculus Examples

$\frac{lim_{x \to 5} 5 x (5 - x)}{3 x - 15}$

Step 1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 5} x (5 - x)}{3 x - 15}$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{5 lim_{x \to 5} x \cdot lim_{x \to 5} 5 - x}{3 x - 15}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{5 lim_{x \to 5} x \cdot (lim_{x \to 5} 5 - lim_{x \to 5} x)}{3 x - 15}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{5 lim_{x \to 5} x \cdot (5 - lim_{x \to 5} x)}{3 x - 15}$

Step 5

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5 \cdot 5 \cdot (5 - lim_{x \to 5} x)}{3 x - 15}$

Step 5.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5 \cdot 5 \cdot (5 - 5)}{3 x - 15}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $5$ by $5$ .

$\frac{25 (5 - 5)}{3 x - 15}$

Step 6.2

Factor $3$ out of $3 x - 15$ .

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Step 6.2.1

Factor $3$ out of $3 x$ .

$\frac{25 (5 - 5)}{3 (x) - 15}$

Step 6.2.2

Factor $3$ out of $- 15$ .

$\frac{25 (5 - 5)}{3 x + 3 \cdot - 5}$

Step 6.2.3

Factor $3$ out of $3 x + 3 \cdot - 5$ .

$\frac{25 (5 - 5)}{3 (x - 5)}$

Step 6.3

Subtract $5$ from $5$ .

$\frac{25 \cdot 0}{3 (x - 5)}$

Step 6.4

Multiply $25$ by $0$ .

$\frac{0}{3 (x - 5)}$

Step 6.5

Cancel the common factor of $0$ and $3$ .

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Step 6.5.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3 (x - 5)}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Cancel the common factor.

$\frac{3 \cdot 0}{3 (x - 5)}$

Step 6.5.2.2

Rewrite the expression.

$\frac{0}{x - 5}$

Step 6.6

Divide $0$ by $x - 5$ .

$0$