Calculus Examples

$\frac{lim_{x \to 9} \frac{5}{x} - \frac{5}{9}}{x - 9}$

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 $9$ .

$\frac{lim_{x \to 9} \frac{5}{x} - lim_{x \to 9} \frac{5}{9}}{x - 9}$

Step 1.2

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

$\frac{5 lim_{x \to 9} \frac{1}{x} - lim_{x \to 9} \frac{5}{9}}{x - 9}$

Step 1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $9$ .

$\frac{5 \frac{lim_{x \to 9} 1}{lim_{x \to 9} x} - lim_{x \to 9} \frac{5}{9}}{x - 9}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $9$ .

$\frac{5 \frac{1}{lim_{x \to 9} x} - lim_{x \to 9} \frac{5}{9}}{x - 9}$

Step 1.5

Evaluate the limit of $\frac{5}{9}$ which is constant as $x$ approaches $9$ .

$\frac{5 \frac{1}{lim_{x \to 9} x} - \frac{5}{9}}{x - 9}$

Step 2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{5 (\frac{1}{9}) - \frac{5}{9}}{x - 9}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $9$ .

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Step 3.1.1

Multiply $\frac{5 (\frac{1}{9}) - \frac{5}{9}}{x - 9}$ by $\frac{9}{9}$ .

$\frac{9}{9} \cdot \frac{5 (\frac{1}{9}) - \frac{5}{9}}{x - 9}$

Step 3.1.2

Combine.

$\frac{9 (5 (\frac{1}{9}) - \frac{5}{9})}{9 (x - 9)}$

Step 3.2

Apply the distributive property.

$\frac{9 (5 (\frac{1}{9})) + 9 (- \frac{5}{9})}{9 x + 9 \cdot - 9}$

Step 3.3

Cancel the common factor of $9$ .

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Step 3.3.1

Move the leading negative in $- \frac{5}{9}$ into the numerator.

$\frac{9 (5 (\frac{1}{9})) + 9 (\frac{- 5}{9})}{9 x + 9 \cdot - 9}$

Step 3.3.2

Cancel the common factor.

$\frac{9 (5 (\frac{1}{9})) + 9 (\frac{- 5}{9})}{9 x + 9 \cdot - 9}$

Step 3.3.3

Rewrite the expression.

$\frac{9 (5 (\frac{1}{9})) - 5}{9 x + 9 \cdot - 9}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $9$ by $5$ .

$\frac{45 (\frac{1}{9}) - 5}{9 x + 9 \cdot - 9}$

Step 3.4.2

Cancel the common factor of $9$ .

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Step 3.4.2.1

Factor $9$ out of $45$ .

$\frac{9 (5) \frac{1}{9} - 5}{9 x + 9 \cdot - 9}$

Step 3.4.2.2

Cancel the common factor.

$\frac{9 \cdot 5 \frac{1}{9} - 5}{9 x + 9 \cdot - 9}$

Step 3.4.2.3

Rewrite the expression.

$\frac{5 - 5}{9 x + 9 \cdot - 9}$

Step 3.4.3

Subtract $5$ from $5$ .

$\frac{0}{9 x + 9 \cdot - 9}$

Step 3.5

Factor $9$ out of $9 x + 9 \cdot - 9$ .

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Step 3.5.1

Factor $9$ out of $9 x$ .

$\frac{0}{9 (x) + 9 \cdot - 9}$

Step 3.5.2

Factor $9$ out of $9 \cdot - 9$ .

$\frac{0}{9 (x) + 9 (- 9)}$

Step 3.5.3

Factor $9$ out of $9 (x) + 9 (- 9)$ .

$\frac{0}{9 (x - 9)}$

Step 3.6

Cancel the common factor of $0$ and $9$ .

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Step 3.6.1

Factor $9$ out of $0$ .

$\frac{9 (0)}{9 (x - 9)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Cancel the common factor.

$\frac{9 \cdot 0}{9 (x - 9)}$

Step 3.6.2.2

Rewrite the expression.

$\frac{0}{x - 9}$

Step 3.7

Divide $0$ by $x - 9$ .

$0$