The example problem that we are going to use so as to show that the Limit of a function does not exists, is ... Find the limit of the following function of x, as x approaches 0; that is,..f(x) = [abs(x)]/x.
Limf(x) = Lim(abs(x))/x, as x->0. (where abs(x), means the absolute value of x.).Please click on the image to see the graph.
We should note, that if we directly substitute the number,0, into the function, f(x)= (abs(x))/x, we get the Indeterminate form of, 0/0.
By the definition of 'the limit of a function as, x, approaches 0', means, x approaching 0 from the left of 0, and x approaching 0, from the right of 0, and x is not equal to 0.
We should then substitute, a few x-numbers that are close to zero and are approaching 0 from the left, into the function.
Let us choose the numbers, -3,-2,-1, and -0.5, when we do the direct substitution, we get f(x) = -1 for each x-number substituted.
Similarly, if we choose a few x-numbers, approaching 0, from the right, that is 3,2,1, and 0.5, we get f(x) = 1. Since the value of f(x) is different -1 and 1, as x approaches 0 from the left and from the right, we say the Limit of the function does NOT EXISTS.
The example problem that we are going to use to show that the limit of a function is Infinity, is,...Find the Limit of the following function of x, as x approaches 0. That is, let f(x) = 1/x, then
Limf(x) = Lim(1/x),as x->0.
Please click on the image to see the graph.
By directly substituting the number 0 for x, in the function,
Limf(x) = Lim(1/x), we get 1/0, which is Undefined, (Any number other than zero, divided by zero is undefined.)
We are going to substitute some x-numbers approaching 0 from the left and also some x-numbers approaching 0 from the right and see the behaviour of f(x).
Let us choose the x-numbers from the left of zero, to be -3,-2,-1,-0.5, and -0.01 and the x-numbers from the right of zero to be 3,2,1,0.5, and 0.01, after direct substitution of the x-numbers to the left of zero into the funtion f(x), we notice that f(x) is approaching Negative Infinity as x->0 from the left of 0, and similarly, f(x) is approaching Positive Infinity as the x-numbers from the right of zero are being substituted into the function of f(x).
Since f(x) is approaching Infinity, as x approaches 0, from the left and from the right, we say the Limit of f(x) is Infinity. But notice that, in this case, f(x) is approaching two different Infinities, so we can safely say the Limit does not exists. In the case that f(x) is approaching Positive Infinity for x approaching 0 both from the left and from the right, Please try the Lim(1/x^2), as x->0, we still say the limit of f(x) is Infinity, and technically speaking the Limit does not exists.