Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Cancel the common factor of $0 - 1 \cdot 1$ and $0 + 1$ .

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Step 8.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{- 1 (0) - 1 \cdot 1}{0 + 1}$

Step 8.2

Factor $- 1$ out of $- 1 \cdot 1$ .

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

Step 8.3

Factor $- 1$ out of $- 1 (0) - 1 (1)$ .

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

Step 8.4

Cancel the common factor.

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

Step 8.5

Divide $- 1$ by $1$ .

$- 1$