Calculus Examples

$f (x) = {\begin{cases} - 3 x - 1 & x < - 1 \\ - x + 1 & x \geq - 1 \end{cases}$

Step 1

Find the limit of $- 3 x - 1$ as $x$ approaches $- 1$ from the left.

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Step 1.1

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

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

Step 1.2

Evaluate the limit.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$- 3 \cdot - 1 - 1 \cdot 1$

Step 1.4

Simplify the answer.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $- 3$ by $- 1$ .

$3 - 1 \cdot 1$

Step 1.4.1.2

Multiply $- 1$ by $1$ .

$3 - 1$

Step 1.4.2

Subtract $1$ from $3$ .

$2$

Step 2

Evaluate $- x + 1$ at $- 1$ .

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Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

$- (- 1) + 1$

Step 2.2

Evaluate.

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Step 2.2.1

Multiply $- 1$ by $- 1$ .

$1 + 1$

Step 2.2.2

Add $1$ and $1$ .

$2$

Step 3

Since the limit of $- 3 x - 1$ as $x$ approaches $- 1$ from the left is equal to the function value at $x = - 1$ , the function is continuous at $x = - 1$ .

Continuous

Step 4