Calculus Examples

$lim_{x \to 3} \frac{1}{\frac{x}{{(x - 3)}^{3}}} - \frac{1}{27}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$lim_{x \to 3} \frac{1}{\frac{x}{{(x - 3)}^{3}}} - lim_{x \to 3} \frac{1}{27}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 3} 1 \frac{{(x - 3)}^{3}}{x} - lim_{x \to 3} \frac{1}{27}$

Step 2.2

Multiply $\frac{{(x - 3)}^{3}}{x}$ by $1$ .

$lim_{x \to 3} \frac{{(x - 3)}^{3}}{x} - lim_{x \to 3} \frac{1}{27}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} {(x - 3)}^{3}}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{27}$

Step 4

Move the exponent $3$ from ${(x - 3)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 3} x - 3)}^{3}}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{27}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - lim_{x \to 3} 3)}^{3}}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{27}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - 1 \cdot 3)}^{3}}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{27}$

Step 7

Evaluate the limit of $\frac{1}{27}$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - 1 \cdot 3)}^{3}}{lim_{x \to 3} x} - \frac{1}{27}$

Step 8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{{(3 - 1 \cdot 3)}^{3}}{lim_{x \to 3} x} - \frac{1}{27}$

Step 8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{{(3 - 1 \cdot 3)}^{3}}{3} - \frac{1}{27}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $- 1$ by $3$ .

$\frac{{(3 - 3)}^{3}}{3} - \frac{1}{27}$

Step 9.2

Subtract $3$ from $3$ .

$\frac{0^{3}}{3} - \frac{1}{27}$

Step 9.3

Simplify each term.

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Step 9.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{3} - \frac{1}{27}$

Step 9.3.2

Divide $0$ by $3$ .

$0 - \frac{1}{27}$

Step 9.4

Subtract $\frac{1}{27}$ from $0$ .

$- \frac{1}{27}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{27}$

Decimal Form:

$- 0.\overline{037}$