Calculus Examples

$lim_{x \to 0} \frac{1}{x - 3}$

Step 1

Change the two-sided limit into a left sided limit.

$lim_{x \to 0^{-}} \frac{1}{x - 3}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to 0^{-}} 1}{lim_{x \to 0^{-}} x - 3}$

Step 2.2

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

$\frac{1}{lim_{x \to 0^{-}} x - 3}$

Step 2.3

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

$\frac{1}{lim_{x \to 0^{-}} x - lim_{x \to 0^{-}} 3}$

Step 2.4

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{1}{lim_{x \to 0^{-}} x - 1 \cdot 3}$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $- 1$ by $3$ .

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

Step 4.1.2

Subtract $3$ from $0$ .

$\frac{1}{- 3}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{1}{3}$