If any is not met, the limit is said to fail to exist or just "does not exist".
If all these conditions are met, the limit is said to exist. This means that you get the SAME value whether x is less than c and increasing toward c OR x is greater than c and and decreasing toward c. Hope that helps :)ĭo you mean a rigorous mathematical definition or an explanation in simpler terms?Ī limit is said to exist for some function f(x) for some value c if f(x) clearly gets closer and closer to some finite value as x gets closer and closer to c. But since in either case (endpoint value included or not) you can keep getting as close as you want to it, the limit is the same. With limits we are saying that no matter how close you want to get to the limit value, you can ALWAYS get closer - it doesn't matter if the limit value endpoint is included or not, you will never 'get' to it anyway since you can always half the distance your are from it. If you tell me you are x distance away from something, I can half the distance and be even closer than you, NO MATTER WHAT DISTANCE x you choose. Now in our physical world, that doesn't happen (and why is another discussion) BUT in the abstract math world, THIS IS TRUE. Zeno argued that motion was impossible since, for example if you shot an arrow at a target the arrow would first have to travel half the distance to the target, then half the remaining distance, the half again and again and again, ad infinitum and thus never reach the target since the arrow always has to traverse the remaining half distance. I hope this helps you understand discontinuities outside of piece-wise functions!Īre you familiar with Zeno's (ancient Greek philosopher) paradox? I find it useful to explain this concept of limits. These discontinuities are called jump discontinuities because you jump from one place to another. However, when it's supposed to get to 1, it jumps down to 0 every time. This means that x-⌊x⌋ is approaching x-(x-1), or 1. From the left side of integers, the numbers will be close to an integer, but the ⌊x⌋ of x will be all the way at the last integer. This is because the distance between two consecutive integers is 1. However, from the left side, the function will be approaching 1.
Let's say that ⌊x⌋ is the greatest integer less than or equal to x. This is a removable discontinuity because all we did was remove one point from the graph and let the graph be normal everywhere else. This is like the piece-wise function y=1 for x does not equal 0 (x/x=1) and y is undefined for x=0 (0/0 is indeterminate) (put your mouse art x=0 and see what y is). This is called an infinite discontinuity. Infinity while from the other side, the graph goes up to Infinity. Many graphs that aren't piece-wise functions have discontinuities.
0 kommentar(er)
