Calculus Examples

$f (x) = {\begin{cases} x - 3 & x \leq 2 \\ - 5 x + 4 & x > 2 \end{cases}$

Step 1

Find the limit of $- 5 x + 4$ as $x$ approaches $2$ from the right.

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Step 1.1

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

$lim_{x \to 2^{+}} - 5 x + 4$

Step 1.2

Evaluate the limit.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$- lim_{x \to 2^{+}} 5 x + lim_{x \to 2^{+}} 4$

Step 1.2.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$- 5 lim_{x \to 2^{+}} x + lim_{x \to 2^{+}} 4$

Step 1.2.3

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$- 5 lim_{x \to 2^{+}} x + 4$

Step 1.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$- 5 \cdot 2 + 4$

Step 1.4

Simplify the answer.

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Step 1.4.1

Multiply $- 5$ by $2$ .

$- 10 + 4$

Step 1.4.2

Add $- 10$ and $4$ .

$- 6$

Step 2

Evaluate $x - 3$ at $2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

$(2) - 3$

Step 2.2

Subtract $3$ from $2$ .

$- 1$

Step 3

Since the limit of $- 5 x + 4$ as $x$ approaches $2$ from the right is not equal to the function value at $x = 2$ , the function is not continuous at $x = 2$ .

Not continuous

Step 4