Question

(a): Does the limit exist? Let

$f\left(x\right)=\left\{\begin{array}{l}x-1\text{if}x<2\\ x^2-4x+6\text{if}x\ge 2\end{array}\right.$

Find the $\underset{x\to 2^{-}}{\lim }f\left(x\right)$and $\underset{x\to 2^{+}}{\lim }f\left(x\right)$

Short answer

The value of$\underset{x\to 2^{-}}{\lim }f\left(x\right)=1$and$\underset{x\to 2^{+}}{\lim }f\left(x\right)=2$

Step-by-step solution

Step 1. Write the given definition.

The expression is

$f\left(x\right)=\left\{\begin{array}{l}x-1\text{if}x<2\\ x^2-4x+6\text{if}x\ge 2\end{array}\right.$

Step 2. Find the limit of the function from the left.

As x approaches 2 from the left,

$\begin{array}{c}\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }(x-1)\\ =2-1\\ =1\end{array}$

Step 3. Find the limit of the function from the right.

As x approaches 2 from the right,

$\begin{array}{c}\underset{x\to 2+}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }(x^2-4x+6)\\ =4-8+6\\ =2\end{array}$

Therefore, value of $\underset{x\to 2^{-}}{\lim }f\left(x\right)=1$and$\underset{x\to 2+}{\lim }f\left(x\right)=2$

 (b): Does the limit exist? Let f(x)=x−1                   if x<2x2−4x+6         if x≥2Doeslimx→2f(x)  exists.

The limit of does not exist.

Step 1. Write the given definition.

The expression is

$f\left(x\right)=\left\{\begin{array}{l}x-1\text{if}x<2\\ x^2-4x+6\text{if}x\ge 2\end{array}\right.$

Step 2. Find the limit of the function from the left.

As $x$approaches $2$from the left,

$\begin{array}{c}\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }(x-1)\\ =2-1\\ =1\end{array}$

Step 3. Find the limit of the function from the right.

As $x$approaches $2$from the right,

$\begin{array}{c}\underset{x\to 2+}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }(x^2-4x+6)\\ =4-8+6\\ =2\end{array}$

Since the limits of the absolute value functions as$x$approaches$2$from both sides are not equal, the limit does not exist.

Therefore, the limit of function does not exist.

Does the limit exist? Let f(x)=x−1                   if x<2x2−4x+6         if x≥2Sketch the graph of f .

The graph of is shown below:

In the graph, red curve represents $f\left(x\right)=x^2-4x+6;x\ge 2$and blue line represents $f\left(x\right)=x-1;x<2$