Calculus Examples

$f (x) = {\begin{cases} - x^{2} + 10 & - 4 \leq x < 4 \\ 6 - 3 x & x \geq 4 \end{cases}$

Step 1

Find the limit of $- x^{2} + 10$ as $x$ approaches $4$ from the left.

Tap for more steps...

Step 1.1

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

$lim_{x \to 4^{-}} - x^{2} + 10$

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 $4$ .

$- lim_{x \to 4^{-}} x^{2} + lim_{x \to 4^{-}} 10$

Step 1.2.2

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

$- {(lim_{x \to 4^{-}} x)}^{2} + lim_{x \to 4^{-}} 10$

Step 1.2.3

Evaluate the limit of $10$ which is constant as $x$ approaches $4$ .

$- {(lim_{x \to 4^{-}} x)}^{2} + 10$

Step 1.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$- 4^{2} + 10$

Step 1.4

Simplify the answer.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

Raise $4$ to the power of $2$ .

$- 1 \cdot 16 + 10$

Step 1.4.1.2

Multiply $- 1$ by $16$ .

$- 16 + 10$

Step 1.4.2

Add $- 16$ and $10$ .

$- 6$

Step 2

Evaluate $6 - 3 x$ at $4$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $4$ in the expression.

$6 - 3 \cdot 4$

Step 2.2

Evaluate.

Tap for more steps...

Step 2.2.1

Multiply $- 3$ by $4$ .

$6 - 12$

Step 2.2.2

Subtract $12$ from $6$ .

$- 6$

Step 3

Since the limit of $- x^{2} + 10$ as $x$ approaches $4$ from the left is equal to the function value at $x = 4$ , the function is continuous at $x = 4$ .

Continuous

Step 4