Calculus Examples

$lim_{x \to \frac{1}{3}} \frac{- 9 x^{2}}{2 - 3 x}$

Step 1

Move the term $- 9$ outside of the limit because it is constant with respect to $x$ .

$- 9 lim_{x \to \frac{1}{3}} \frac{x^{2}}{2 - 3 x}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$- 9 \frac{lim_{x \to \frac{1}{3}} x^{2}}{lim_{x \to \frac{1}{3}} 2 - 3 x}$

Step 3

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

$- 9 \frac{{(lim_{x \to \frac{1}{3}} x)}^{2}}{lim_{x \to \frac{1}{3}} 2 - 3 x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$- 9 \frac{{(lim_{x \to \frac{1}{3}} x)}^{2}}{lim_{x \to \frac{1}{3}} 2 - lim_{x \to \frac{1}{3}} 3 x}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{1}{3}$ .

$- 9 \frac{{(lim_{x \to \frac{1}{3}} x)}^{2}}{2 - lim_{x \to \frac{1}{3}} 3 x}$

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- 9 \frac{{(lim_{x \to \frac{1}{3}} x)}^{2}}{2 - 3 lim_{x \to \frac{1}{3}} x}$

Step 7

Evaluate the limits by plugging in $\frac{1}{3}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$- 9 \frac{{(\frac{1}{3})}^{2}}{2 - 3 lim_{x \to \frac{1}{3}} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$- 9 \frac{{(\frac{1}{3})}^{2}}{2 - 3 (\frac{1}{3})}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the product rule to $\frac{1}{3}$ .

$- 9 \frac{\frac{1^{2}}{3^{2}}}{2 - 3 (\frac{1}{3})}$

Step 8.1.2

One to any power is one.

$- 9 \frac{\frac{1}{3^{2}}}{2 - 3 (\frac{1}{3})}$

Step 8.1.3

Raise $3$ to the power of $2$ .

$- 9 \frac{\frac{1}{9}}{2 - 3 (\frac{1}{3})}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Factor $3$ out of $- 3$ .

$- 9 \frac{\frac{1}{9}}{2 + 3 (- 1) \frac{1}{3}}$

Step 8.2.1.2

Cancel the common factor.

$- 9 \frac{\frac{1}{9}}{2 + 3 \cdot - 1 \frac{1}{3}}$

Step 8.2.1.3

Rewrite the expression.

$- 9 \frac{\frac{1}{9}}{2 - 1}$

Step 8.2.2

Subtract $1$ from $2$ .

$- 9 \frac{\frac{1}{9}}{1}$

Step 8.3

Divide $\frac{1}{9}$ by $1$ .

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

Step 8.4

Cancel the common factor of $9$ .

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Step 8.4.1

Factor $9$ out of $- 9$ .

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

Step 8.4.2

Cancel the common factor.

$9 \cdot - 1 \frac{1}{9}$

Step 8.4.3

Rewrite the expression.

$- 1$