Example 1: Continuity Using an Equation

Determine if this function is continuous at $\mathit{x}=3$.

$f\left(\mathrm{(}x\mathrm{)}\right)=\left\{\{\begin{array}{c}x+2,\mathit{ }x<3 \\ 5,\mathit{ }x=3\\ 3x–4,\mathit{ x}>3\end{array}\right.$

Step 1: Determine the overall limit, $\underset{x\to c}{\lim } f\left(x\right)$, at your given x-value .

Here we are only looking at $\mathit{x}=3$.

We break the overall limit into two limits, LHL and RHL.

When you work with a piecewise function the hard part is determining which piece of the function to choose.

For the LHL we want values to the left of $\mathit{x}=3$. That would mean when $x<3$, so we would choose the top equation, $x+2$.

For the RHL we want values to the right of $\mathit{x}=3$. That would mean when $x>3$, so we would choose the bottom equation, $3x–4$.

In this example we determine that LHL = RHL, and therefore, the overall limit is the same value, $\underset{x\to 3}{\lim } f\left(x\right)=5$.

$\underset{x\to 3}{\lim } f\left(x\right)=$

Left-Hand Limit

$\underset{x\to 3^{–}}{\lim }x+2=\left(3\right)+2=5$

Right-Hand Limit

$\underset{x\to 3^{+}}{\lim }3x–4=3\left(3\right)–4=5$

LHL = RHL

5 = 5

$\underset{x\to 3}{\lim } f\left(x\right)=5$

Step 2: Determine the function value, $f\left(c\right)$, at your given x-value .

Here we would need to determine $f\left(3\right)$. Since we need to know what the value is when $x=3$, which means we would use the middle equation which happens to just be a constant, 5.

$f\left(3\right)=5$

Step 3: Determine if the overall limit equals the function value, $\underset{x\to c}{\lim } f\left(x\right)=f\left(c\right)$.

Here we found that the limit, $\underset{x\to 3}{\lim } f\left(x\right)=5$, did equal the function value, $f\left(3\right)=5$.

Or as I say, the limit equaled the dot on the graph.

$\underset{x\to 3}{\lim } f\left(x\right)=5=f\left(3\right)=5$

$\underset{x\to 3}{\lim } f\left(x\right)=f\left(3\right)$

Final Result:

The function, $f\left(x\right)$, is continuous at x = 3 because $\underset{x\to 3}{\lim } f\left(x\right)=f\left(3\right)$.