Calculus Examples

$lim_{x \to 0} (\frac{1}{x + 6}) - \frac{1}{\frac{6}{x}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \frac{1}{x + 6} - lim_{x \to 0} \frac{1}{\frac{6}{x}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 1}{lim_{x \to 0} x + 6} - lim_{x \to 0} \frac{1}{\frac{6}{x}}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{lim_{x \to 0} x + 6} - lim_{x \to 0} \frac{1}{\frac{6}{x}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{lim_{x \to 0} x + lim_{x \to 0} 6} - lim_{x \to 0} \frac{1}{\frac{6}{x}}$

Step 5

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$\frac{1}{lim_{x \to 0} x + 6} - lim_{x \to 0} \frac{1}{\frac{6}{x}}$

Step 6

Simplify the limit argument.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{lim_{x \to 0} x + 6} - lim_{x \to 0} 1 \frac{x}{6}$

Step 6.2

Multiply $\frac{x}{6}$ by $1$ .

$\frac{1}{lim_{x \to 0} x + 6} - lim_{x \to 0} \frac{x}{6}$

Step 7

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{lim_{x \to 0} x + 6} - \frac{1}{6} lim_{x \to 0} x$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{0 + 6} - \frac{1}{6} lim_{x \to 0} x$

Step 8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{0 + 6} - \frac{1}{6} \cdot 0$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Add $0$ and $6$ .

$\frac{1}{6} - \frac{1}{6} \cdot 0$

Step 9.1.2

Multiply $- \frac{1}{6} \cdot 0$ .

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Step 9.1.2.1

Multiply $0$ by $- 1$ .

$\frac{1}{6} + 0 (\frac{1}{6})$

Step 9.1.2.2

Multiply $0$ by $\frac{1}{6}$ .

$\frac{1}{6} + 0$

Step 9.2

Add $\frac{1}{6}$ and $0$ .

$\frac{1}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$