Limits: Introduction and OneSided Limits
$\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L$
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 $f\left(x\right)=\frac{x+2}{x1}$ as it approaches 1. The table lists the values of f(x) near x = 0.
x 
$f\left(x\right)=\frac{x+2}{x1}$ 
x 
$f\left(x\right)=\frac{x+2}{x1}$ 
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 
$\underset{x\to 0}{\mathrm{lim}}\frac{\left(x+2\right)}{x1}=2$
While the limit of the function $f\left(x\right)=\frac{x+2}{x1}$ seems to approach 2 as x approaches 0 from either the left or the right, some function have only onesided limits. The following notation is used to denoted lefthand and righthand limits.
$\underset{x\to {a}^{}}{\mathrm{lim}}f\left(x\right)$$\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$
Lefthand limitRighthand 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 lefthand and righthand 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.
Lefthand Limit: $\underset{x\to {1}^{}}{\mathrm{lim}}f\left(x\right)=3$
Righthand 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 $x\to 2$
Lefthand Limit: $\underset{x\to {2}^{}}{\mathrm{lim}}f\left(x\right)=0.5$
Righthand 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.
$\underset{x\to 2}{\mathrm{lim}}\frac{\sqrt{x1}+3}{x}$
Step 1: Graph the function.


Step 2: Create a table of values close to and on either side of 2.
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. $\underset{x\to 2}{\mathrm{lim}}f\left(x\right)=2$. 
$\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{6}+2}{{x}^{8}+1}$
Step 1: Graph the function.


Step 2: Create a table for values close to and on either side of 1.
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. $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)=1.5$. 
$\underset{x\to {3}^{}}{\mathrm{lim}}f\left(x\right)$ $\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)$ $\underset{x\to 3}{\mathrm{lim}}f\left(x\right)$ $f\left(3\right)$ $\underset{x\to {1}^{}}{\mathrm{lim}}f\left(x\right)$ $\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)$ $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)$
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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