Calculus Examples

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

Step 1

Find the limit of $4 - x^{2}$ as $x$ approaches $1$ .

Tap for more steps...

Step 1.1

Evaluate the limit.

Tap for more steps...

Step 1.1.1

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

$lim_{x \to 1} 4 - lim_{x \to 1} x^{2}$

Step 1.1.2

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

$4 - lim_{x \to 1} x^{2}$

Step 1.1.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$4 - {(lim_{x \to 1} x)}^{2}$

Step 1.2

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

$4 - 1^{2}$

Step 1.3

Simplify the answer.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

One to any power is one.

$4 - 1 \cdot 1$

Step 1.3.1.2

Multiply $- 1$ by $1$ .

$4 - 1$

Step 1.3.2

Subtract $1$ from $4$ .

$3$

Step 2

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

Tap for more steps...

Step 2.1

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

$2 (1) + 1$

Step 2.2

Evaluate.

Tap for more steps...

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 $4 - x^{2}$ as $x$ approaches $1$ is equal to the function value at $x = 1$ , the function is continuous at $x = 1$ .

Continuous

Step 4