Calculus Examples

$f (x) = {\begin{cases} - x + 4 & x < 1 \\ 2 x + 1 & x \geq 1 \end{cases}$

Step 1

Find the limit of $- x + 4$ as $x$ approaches $1$ from the left.

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Step 1.1

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

$lim_{x \to 1^{-}} - x + 4$

Step 1.2

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

$- lim_{x \to 1^{-}} x + lim_{x \to 1^{-}} 4$

Step 1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$- lim_{x \to 1^{-}} x + 4$

Step 1.4

Simplify the expression.

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Step 1.4.1

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

$- 1 + 4$

Step 1.4.2

Add $- 1$ and $4$ .

$3$

Step 2

Evaluate $2 x + 1$ at $1$ .

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Step 2.1

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

$2 (1) + 1$

Step 2.2

Evaluate.

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Step 2.2.1

Multiply $2$ by $1$ .

$2 + 1$

Step 2.2.2

Add $2$ and $1$ .

$3$

Step 3

Since the limit of $- x + 4$ 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