What is the limit of 1divided by the absolute value of x?

Answer 1

If you plot the graph of f(x) = 1/|x| in your graphing calculator and observe it, you'll see that the graph has a break in it (the line x = 0 (y-axis) is a vertical asymptote of the graph of the function), and is composed by two branches, which lie above the x-axis (because of the absolute value, and the line y = 0 is a horizontal asymptote).

So that,

as x approaches 0+ (from the right), f(x) increases without bound: lim(x→0+) 1/|x| = ∞, and
as x increases without bound, the values of f(x) approaches 0: lim(x→+∞) 1/|x| = 0
as x approaches 0- (from the left), f(x) increases without bound: lim(x→0+) 1/|x| = ∞, and
as x decreases without bound, the values of f(x) approaches 0: lim(x→-∞) 1/|x| = 0

Thus, lim(x→± ∞) 1/|x| = 0.

By taking x close enough to 0, the values of f(x) do not approaches a number, so lim(x→0) 1/|x| does not exist (when we write symbolically lim(x→0) 1/|x| = ∞, we simply express the particular way in which the limit does not exist; 1/|x| can be made as large as possible as we like by taking x close enough to 0, but not equal to 0).