This method is generally only used to express the notion of a limit, and not to compute limits, as it is quite a lengthy one:
For example, there is a function `f(x) = 2x^2 + 3x + 1`. In order to find its left and right limits as x approaches 1, you need to proceed as follows:
It is clear that the function values are approaching 6 as x is approaching 1. Thus the left limit of `f(x) = 2x^2 + 3x + 1` as x approaches 1 is 6. This is written as `lim_(x > 1^) f(x) = 2x^2 + 3x + 1`
Similarly, the right limit is obtained by taking an x value just to the right of x = 1 and then moving closer and closer towards x = 1 from it. Evaluating the function at each x value, we get the following table for the right limit:
The function values are approaching 6 as x is approaching 1 from the right hand side. So the right hand limit of the function as x approaches x is 6, `lim_(x > 1^+) f(x) = 6`
For example, there is a function `f(x) = 2x^2 + 3x + 1`. In order to find its left and right limits as x approaches 1, you need to proceed as follows:
Left limit as x approaches 1:
A limit is talking about the general value which the function is approaching as x approaches a certain value. In hes 1. Its left limit is the general value which the function takes to the left of x = 1. Thus, in order to see where the function is going without a graph, we assume an x value to the left of 1 (smaller than 1), say x = 0.1, and move closer to 1 in small steps of, say, 0.01. Evaluating the function at each of these x values, we get the following table:
x value

Function
value

0.90

f(0.90) =
5.32

0.91

f(0.91) =
5.3862

0.92

f(0.92) =
5.4528

0.93

f(0.93) =
5.5198

0.94

f(0.94) =
5.5872

0.95

f(0.95) =
5.655

0.96

f(0.96) =
5.7232

0.97

f(0.97) =
5.7918

0.98

f(0.98) =
5.8608

0.99

f(0.99) =
5.9302

1.00

f(1.00) =
6.0

Left limit as x approaches 1
It is clear that the function values are approaching 6 as x is approaching 1. Thus the left limit of `f(x) = 2x^2 + 3x + 1` as x approaches 1 is 6. This is written as `lim_(x > 1^) f(x) = 2x^2 + 3x + 1`
Similarly, the right limit is obtained by taking an x value just to the right of x = 1 and then moving closer and closer towards x = 1 from it. Evaluating the function at each x value, we get the following table for the right limit:
Right limit as x approaches 1
x value

Function value

1.10

f(1.1) = 6.72

1.01

f(1.09) = 6.6462

1.02

f(1.08) = 6.5728

1.03

f(1.07) = 6.4998

1.04

f(1.06) = 6.4272

1.05

f(1.05) = 6.355

1.06

f(1.04) = 6.2832

1.07

f(1.03) = 6.2118

1.08

f(1.02) = 6.1408

1.09

f(1.01) = 6.0702

1.00

f(1.00) = 6.0

Right limit as x approaches 1
Normal limit as x approaches 1
Since both the left and right limits are equal, then the normal limit of the function is equal to them. So the normal limit of the function as x is approaching 1 is 6. That is `lim_(x > 1) f(x) = 6`
The above method of finding the limits of a function is used to demonstrate the definition of the limits of a function for any value of x
The limit of a function as x approaches a particular value is the value that the function is approaching around x.
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