Calculus Examples

$f (x) = {\begin{cases} x + 2 & x < 0 \\ e^{x} & 0 \leq x \leq 1 \\ 2 - x & x > 1 \end{cases}$

Step 1

Find the limit of $x + 2$ as $x$ approaches $0$ from the left.

Tap for more steps...

Step 1.1

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

$lim_{x \to 0^{-}} x + 2$

Step 1.2

Evaluate the limit.

Tap for more steps...

Step 1.2.1

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

$lim_{x \to 0^{-}} x + lim_{x \to 0^{-}} 2$

Step 1.2.2

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$lim_{x \to 0^{-}} x + 2$

Step 1.3

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

$0 + 2$

Step 1.4

Add $0$ and $2$ .

$2$

Step 2

Evaluate $e^{x}$ at $0$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $0$ in the expression.

$e^{0}$

Step 2.2

Anything raised to $0$ is $1$ .

$1$

Step 3

Since the limit of $x + 2$ as $x$ approaches $0$ from the left is not equal to the function value at $x = 0$ , the function is not continuous at $x = 0$ .

Not continuous

Step 4