Calculus: Limits

Question

What makes a limit become non existence?

I know that on a graph, you can tell if a limit exists or not when a^- and a^+ are equal. But if I am finding the limit algebraically or by using a table, how would I know?

Answer 1

1. You can determine whether a limit is existent or not by testing the limit algebraically. For example, you can determine this by looking at your left and right limits and you'd typically see this as a piecewise function. Ex.) f(x) = 3x + 1, if x < 2 and f(x) = x² - 4, if x >2. So to solve for the limit approaching from the left, we use lim_x->2-f(x)= 3x + 1. We plug in 2 to this function and lim_x->2-f(x) = 7. Now do the same thing for the limit approaching from the right so, lim _x->2+f(x) = x² - 4, so lim_x->2+f(x) = 0. In this case, the left and right limit do not equal one another so lim_x->2 f(x) = DNE.
2. The other way to test limits without a graph is by a table.
3. Example: f(x) = (x² - 4)/(x-2) and we need to find the limit as x -> 2 and the x values of the table are 1.9, 1.99. 1.999, 2.001, 2.01, 2.1.
4. We can simplify our function by factoring our the numerator so, f(x) = [(x-2)(x+2)]/(x-2), so f(x) = x+2.
5. To determine lim_{x ->2-}, you'd plug in [1.9,1.999] into our simplified function, which gives the limit approaches 4 from the left. To determine the right approaching limit, use [2.001,2.1] to plug into the simplified function (I did it in reverse order because that is the way it approaches x =2). The results in the right approaching limit equal to 4. Since our left and right limits equal four, the limit as x approaches 2 is 4.

Answer 2

Make a table of values.

The limit from one side isn't the same as the limit from the other side. y = |x|/x is -1 from the left, but it's 1 from the right.

x= -.5, y = |-.5|/-.5 = -1

x = .5, y = |.5|/|.5 = 1

You'd see that with many values in a table.

The value fluctuates. For example, y = sin(1/x) goes between positive and negative as it approaches 0:

sin (1/.4) = +

sin(1/.3) = -

sin (1/.04) = +

sin(1/.03) = -

It has a vertical asymptote: y = 1/x has an vertical asymptote at x=0. Any value that makes the denominator 0 or a negative inside an even root creates a vertical asymptote.