What are Limits?
Please explain Limits in absolutely simple langusge without the usage of any technical language. Actually, i am a newbie to Calculus and i know nothing more than what are functions and a little bit avout about ordered pairs in Calculus.
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There are different kinds of limits in calculus --the limit as x approaches c of a function, the limit of a sequence or series, etc... In a typical first semester calculus class you are probably working with `lim_(x->c)f(x)` .
There area large number of technical things to consider, but basically what `lim_(x->c)f(x)=L` says is that as x gets arbitrarily close to c, f(x) gets arbitrarily close to L. (The rub comes when we try to define "arbitrarily close"-- for that we use a `delta-epsi` definition).
If you look at a table of values for f(x), when x is close to c the function value (y=f(x) ) is close to L. The closer x gets to c, in general the closer y gets to L.
If you graph f(x), as you trace along the curve and get closer to c in the x-direction, the y-value is getting closer to L.
You might consider when a function fails to have a limit at a point -- here are some possibilities:
(1) If the function grows without bound as x nears c (e.g. `y=1/x` with c=0) then there is no limit since as x approaches c the y-value does not approach a number L-- it just keeps growing past any number you specify.
(2) If the function experiences a "jump" at c: define f(x) to be x if x>0, and f(x) to be -2x-3 for `x<=0` , and let c=0. As you approach c from the left, f(x) approaches -3; but as you approach 0 from the right, f(x) approaches 0. Note that as x approaches 0, f(x) does not approach a number L; it approaches two different numbers so there is no limit.
(3) Finally, consider a function with infinite oscillations at c. Since the function goes from its maximum to its minimum an infinite number of times, you cannot say that f(x) approaches any single number.
A limit is just what a function approaches as we let x approach a certain value. You might recall determining end behavior of functions in your precalculus class. That is like finding the limit of a function as we let x approach negative or positive infinity. A limit can also be when we let x approach a number. In that case, we want to see what value the function approaches from the left side of that number and from the right side of that number. If the two sides approach the same number, then the limit exists, if not we have a discontinuity.
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