Expert: Abe Mantell - 8/30/2005

Question
hi! i am a highschool senior in calculus b/c and i have a real general question about limits. what is a limit? i am having a real hard time understanding the basic definition and use of a limit in a mathmatical equation. i have asked my teacher numerous times and i am just trying to get a diffrent vantage point. so if you would be so kind as to email me back as soon as possible, so i can finnaly understand the basics of calculus!

Answer
Hello Vince,

The major idea of a limit is what value does a function
"appear" to be approaching, irrespective of whether or
not that value is actually attained. Remember, if we
seek the limit (as x-->a) f(x), it means what value
does f(x) "appear" to be approaching as x gets closer
and closer to a.  f(a) may or may not be defined...
as long as the limit exists!
Examples:
1. limit (as x-->0) x/x is clearly 1, since any non-zero
-- number divided by itself is 1...BUT clearly we cannot
-- just let x=0, since 0/0 is what's called an
-- indeterminate form.

2. limit (as x-->1) (x^2-1)/(x-1) = 2, even though
-- (x^2-1)/(x-1) never actually achieves the value
-- of 2...as x gets closer and closer to 1, the
-- expression gets closer and closer to 2.

I hope this helps...here are some sites to check for
more that may also help:
http://www.coolmath.com/limit1.htm
http://archives.math.utk.edu/visual.calculus/1/
http://www.calculus-help.com/funstuff/phobe.html
http://tutorial.math.lamar.edu/AllBrowsers/2413/Tangents_Rates.asp

Abe