Calculus Examples

f (x) = {\begin{cases} - 3 x + 7 & x \leq 3 \\ - 2 & x > 3 \end{cases}

$f (x) = {\begin{cases} - 3 x + 7 & x \leq 3 \\ - 2 & x > 3 \end{cases}$

Step 1

Find the limit of

- 2

$- 2$ as

x

$x$ approaches

3

$3$ from the right.

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Step 1.1

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

lim_{x \to 3^{+}} - 2

$lim_{x \to 3^{+}} - 2$

Step 1.2

Evaluate the limit of

- 2

$- 2$ which is constant as

x

$x$ approaches

3

$3$ .

- 2

$- 2$

- 2

$- 2$

Step 2

Evaluate

- 3 x + 7

$- 3 x + 7$ at

3

$3$ .

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Step 2.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

- 3 \cdot 3 + 7

$- 3 \cdot 3 + 7$

Step 2.2

Evaluate.

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Step 2.2.1

Multiply

- 3

$- 3$ by

3

$3$ .

- 9 + 7

$- 9 + 7$

Step 2.2.2

Add

- 9

$- 9$ and

7

$7$ .

- 2

$- 2$

- 2

$- 2$

- 2

$- 2$

Step 3

Since the limit of

- 2

$- 2$ as

x

$x$ approaches

3

$3$ from the right is equal to the function value at

x = 3

$x = 3$ , the function is continuous at

x = 3

$x = 3$ .

Continuous

Step 4

f (x) = {\begin{matrix} - 3 x + 7 & x \leq 3 \\ - 2 & x > 3 \end{matrix}

$f (x) = {\begin{matrix} - 3 x + 7 & x \leq 3 \\ - 2 & x > 3 \end{matrix}$

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