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