A function's value can be undefined and still have a limit
Move x infinitely close to some value and see where f(x) heads, then compare three ways a graph can break apart (a hole, a jump, and divergence).
"The limit of a function" sounds difficult, but it's really a simple question: "As x gets infinitely close to some value, where does f(x) head?" This is a separate question from whether the function's value is actually defined at that point โ that's why a limit can exist even at a point where the function isn't defined.
This concept becomes the root of differentiation (instantaneous rate of change) and integration (area), which you learn later. Differentiation is defined as "the limit of squeezing an interval infinitely small," and integration is defined as "the limit of cutting rectangles into infinitely many, infinitely thin pieces." Without the concept of a limit, neither one could even be defined.
Use the slider to move x closer and closer to 1 and see what value f(x) heads toward, then below, directly compare three ways a graph can break apart: a hole, a jump, and divergence.
Limit โ f(x) = (xยฒโ1)/(xโ1) is undefined at x=1, since the denominator becomes 0 there. But at every point other than x=1, it's exactly equal to (xโ1)(x+1)/(xโ1) = x+1, so as you move x infinitely close to 1, f(x) gets infinitely close to 2. This is exactly what it means to say "the limit of f(x) as xโ1 is 2."
x0.50
f(x) = (xยฒโ1) รท (xโ1)
For a function to be continuous, all three of these need to hold: โ f(c) is defined, โก the limit lim(xโc)f(x) exists, and โข the two are equal (f(c)=the limit value). Each of the three examples below breaks this condition for a different reason.