Calculus Examples

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

Step 1

Find the limit of $- 2$ as $x$ approaches $3$ from the right.

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Step 1.1

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

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

Step 1.2

Evaluate the limit of $- 2$ which is constant as $x$ approaches $3$ .

$- 2$

Step 2

Evaluate $- 3 x + 7$ at $3$ .

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Step 2.1

Replace the variable $x$ with $3$ in the expression.

$- 3 \cdot 3 + 7$

Step 2.2

Evaluate.

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Step 2.2.1

Multiply $- 3$ by $3$ .

$- 9 + 7$

Step 2.2.2

Add $- 9$ and $7$ .

$- 2$

Step 3

Since the limit of $- 2$ as $x$ approaches $3$ from the right is equal to the function value at $x = 3$ , the function is continuous at $x = 3$ .

Continuous

Step 4