Calculus Examples

$f (x) = {\begin{cases} - x + 2 & x \leq 1 \\ 5 x - 2 & x > 1 \end{cases}$

Step 1

Find the limit of $5 x - 2$ as $x$ approaches $1$ from the right.

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Step 1.1

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

$lim_{x \to 1^{+}} 5 x - 2$

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 $1$ .

$lim_{x \to 1^{+}} 5 x - lim_{x \to 1^{+}} 2$

Step 1.2.2

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

$5 lim_{x \to 1^{+}} x - lim_{x \to 1^{+}} 2$

Step 1.2.3

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$5 lim_{x \to 1^{+}} x - 1 \cdot 2$

Step 1.3

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

$5 \cdot 1 - 1 \cdot 2$

Step 1.4

Simplify the answer.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $5$ by $1$ .

$5 - 1 \cdot 2$

Step 1.4.1.2

Multiply $- 1$ by $2$ .

$5 - 2$

Step 1.4.2

Subtract $2$ from $5$ .

$3$

Step 2

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

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Step 2.1

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

$- (1) + 2$

Step 2.2

Evaluate.

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Step 2.2.1

Multiply $- 1$ by $1$ .

$- 1 + 2$

Step 2.2.2

Add $- 1$ and $2$ .

$1$

Step 3

Since the limit of $5 x - 2$ as $x$ approaches $1$ from the right is not equal to the function value at $x = 1$ , the function is not continuous at $x = 1$ .

Not continuous

Step 4