# Limits: Introduction and One-Sided Limits

The limit of a function f(x) as x approaches some arbitrary value a is denoted by:

$\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L$

and is read "the limit of f(x) as x approaches a equals L". So as the value of x approaches the value of a, the value of f(x) approaches L. It is important to note that the limit does not include where x = a but only the values close to and on either side of a.

Take the function $f\left(x\right)=\frac{x+2}{x-1}$ as it approaches 1. The table lists the values of f(x) near x = 0.

 x $f\left(x\right)=\frac{x+2}{x-1}$ x $f\left(x\right)=\frac{x+2}{x-1}$ -0.1 -1.72 0.0001 -2.0003 -0.5 -1.8571 0.001 -2.003003 -0.01 -0.1.9702 0.01 -2.030303 -0.001 -1.997003 0.05 -2.157895 -0.0001 -1.9997 0.1 -2.333333
The table shows that as x approaches 0 from either the left or the right, the value of f(x) approaches -2. From this we can guesstimate that the limit of $f\left(x\right)=\frac{x+2}{x-1}$ as x approaches 0 is -2:

$\underset{x\to 0}{\mathrm{lim}}\frac{\left(x+2\right)}{x-1}=-2$

While the limit of the function $f\left(x\right)=\frac{x+2}{x-1}$ seems to approach -2 as x approaches 0 from either the left or the right, some function have only one-sided limits. The following notation is used to denoted left-hand and right-hand limits.

$\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)$$\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$

Left-hand limitRight-hand limit

Let's take a look at some limits of the function graphed below.

The limit as x approaches 1 from the left, $\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)$, is 3 while the limit as x approaches 1 from the right, $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)$, is 1. Since the left-hand and right-hand limit as x approaches 1 are different, the limit as x approaches 1 does not exist.

Now' let's look at the limits as x approaches 2. The limit as x approaches 2 from the left is 0.5, and the limit as x approaches 2 from the right is 0.5. Therefore the limit as x approaches 1 from either direction is 0.5.

Note that the when x = 2 the value of the function is 4. Thus the limit is not concerned about the value when x = 2 but only the value as x appraoches 2.

Limits as $x\to 1$

Left-hand Limit:   $\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)=3$

Right-hand Limit: $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)=1$

Overall Limit:      $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)$ = DNE

Limits as $x\to 2$

Left-hand Limit:  $\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=0.5$

Right-hand Limit: $\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=0.5$

Overall Limit:    $\underset{x\to 2}{\mathrm{lim}}f\left(x\right)$ = 0.5

Let's find the limits in a couple examples.

Example 1: Sketch a graph and create a table to determine the limit of the function as $x\to 2$.

$\underset{x\to 2}{\mathrm{lim}}\frac{\sqrt{x-1}+3}{x}$

Step 1: Graph the function.

Step 2: Create a table of values close to and on either side of 2.

 x $f\left(x\right)=\frac{\sqrt{x-1}+3}{x}$ x $f\left(x\right)=\frac{\sqrt{x-1}+3}{x}$ 1.8 2.1635 2.01 1.9925311 1.9 2.0782 2.05 1.9632659 1.95 2.0382 2.1 1.9280042 1.99 2.0075314 2.2 1.861566

The value as x approaches 2 from both the left and the right approaches 2.

$\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=2$ ; $\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=2$

Step 3: Guesstimate the limit

Since the limit from both the left and the right are the same, then the overall limit as x approaches 2 is 2.

$\underset{x\to 2}{\mathrm{lim}}f\left(x\right)=2$.

Example 2: Sketch a graph and create a table to determine the limit of the function as $x\to 1$.

$\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{6}+2}{{x}^{8}+1}$

Step 1: Graph the function.

Step 2: Create a table for values close to and on either side of 1.

 x $f\left(x\right)=\frac{\sqrt{x-1}+3}{x}$ x $f\left(x\right)=\frac{\sqrt{x-1}+3}{x}$ 0.9 1.7696 1.0001 1.4997 0.99 1.5298 1.001 1.4969 0.999 1.5029 1.01 1.4969 0.9999 1.5003 1.1 1.1997628

The value as x approaches 1 from both the left and the right approaches 1.5.

$\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)=1.5$ ; $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)=1.5$

Step 3: Guesstimate the limit

Since the limit from both the left and the right are the same, then the overall limit as x approaches 1 is 1.5.

$\underset{x\to 1}{\mathrm{lim}}f\left(x\right)=1.5$.

Example 3: For the function whose graph is given state the value of each quantity if it exists. If it does not exist, explain why.

$\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)$     $\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)$     $\underset{x\to 3}{\mathrm{lim}}f\left(x\right)$     $f\left(3\right)$     $\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)$     $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)$     $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)$

 Graph of Function Step 1: Evaluate the limits as x approaches 3. $\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)$ - As x approaches 3 from the left, the value of f(x) approaches 2. $\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)$ - As 3 approaches 3 from the right, the value of f(x) approaches 0. $\underset{x\to 3}{\mathrm{lim}}f\left(x\right)$ - Since the value of f(x) as x approaches 3 from the left does not equal the value as x approaches 3 from the right, this limit does not exist. Step 2: Evaluate f(3). The value of y when x is 3 is -1. f(3) = -1 Notice that the limits of f(x) as $x\to 1$ from the left or right may not be related to the value when x = 1. Step 3: Evaluate the limits as x approaches 1. $\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)$ - As x approaches 1 from the left, the value of f(x) approaches 2. $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)$ - As x approaches 1 from the right, the value of f(x) approaches 2. $\underset{x\to 3}{\mathrm{lim}}f\left(x\right)$ - Since the value of f(x) as x approaches 1 from the left equals the value as x approaches 1 from the right, this limit is also 2.

 Related Links: Math algebra Limits: Infinite Limits Limits: Limit Laws Calculus Topics

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