Finding Limits Numerically

Finding limits numerically which means we have to consider approximately some values of x which are closer to the given number from left and right side. And check for which value of x we are getting the function undefined. We can check this by graphing also.

Steps for sketching the graph

Step 1 : Find the x-intercepts of the given function.
Step 2 : Find the y-intercept of the given function.
Step 3 : Mark if there is any vertical or horizontal asymptotes if any.
Step 4 : Get some x-values, plug in that values in the given function and find the y values. (Make a function table).

Examples on Finding Limits Numerically

Example 1 : Sketch the graph of the function

$f(x) = \frac{x^{3}-1}{x-1}$

Solution : First we will graph the given function using curve sketching technique explained above.
As the denominator contains x-1, so we get confused what to do when x=1. So to get an idea about the behavior of the graph near to x= 1. We will use the two set of values of x, one set of values from left side approaches to 1 and other set of values from the right side approaches to 1.
The graph of the given function is a parabola but there is a gap at (1,3) as shown in the above graph. From the graph and the table will understand that x can not be equal to 1 and the f(x) or y will be close to 3. So we will write this using limit notation as

$\lim_{x->1}f(x) = 3$

This can be read as " the limit of f(x), x approaches to 1 is 3"

Example 2 : Estimate the limit numerically.

$f(x)=\frac{x-4}{x^{2}-3x-4}$

$ \lim_{x->4}\frac{x-4}{x^{2}-3x-4}$

Solution : As the given limit approaches to x= 4 so we will consider some values of x from left side of 4 and some values of x from the right side of 4.
x values from the left side = 3.9,3.99,3.999,4
x values from the right side = 4.001,4.01,4.1
We will plug in the above value in the given function.
x= 3.9
$f(3.9)=\frac{3.9-4}{3.9^{2}-3(3.9)-4} =0.2040 $

x=3.99
$f(3.99)=\frac{3.99-4}{3.99^{2}-3(3.99)-4} =0.200400 $

x=3.999
$f(3.999)=\frac{3.999-4}{3.999^{2}-3(3.999)-4} =0.20004000 $

x= 4
$f(4)=\frac{4-4}{4^{2}-3(4)-4} = undefined (no value)$

x= 4.001
$f(4.001)=\frac{4.001-4}{4.001^{2}-3(4.001)-4} = 1.99960$

x= 4.01
$f(4.01)=\frac{4.01-4}{4.01^{2}-3(4.01)-4} = 1.99601$

x=4.1
$f(4.1)=\frac{4.1-4}{4.1^{2}-3(4.1)-4} = 1.996078$

From the above, we can see that when we get closer to 4 the value gets closer to 0.20. So we can conclude that
$ \lim_{x->4}\frac{x-4}{x^{2}-3x-4} \approx 0.2$

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