Section 2.2
The Limit of a Function
- limits occur when the x value approaches a certain value, f(x) also approaches a certain value
- the general notation for limits: lim x->a f(x) = L,"the limit of f(x), as x approaches a, equals L"
- f(x) can be arbitrarily close to a but will never equal a
- calculators can often make mistakes! (refer to example 2 on pages 94-95) on a calculator, if t is sufficiently small, you will get the value 0.
- when a graph shoes infintely many values for x that approach 0, like on the graph of lim x->0 sin (pi/x). the limit does not exist.
- when guessing a limit using a calculator, you cannot be 100% sure because if you can persevere using smaller and smaller values of x, results may be different.
One-Sided Limits
- Definition: written as lim x-> a- f(x) = L, the left hand limit of f(x) as x appraoches a [or the limit of f(x) as x approaches a from the left] is equal to L if f(x) is arbitrarily close to L, and x sufficiently close but LESS THAN a.
- "the right hand limit of f(x) as x approaches a is equal to L" is written lim x-> a+ f(x) = L, and in this case we are only considering when x>a. this leads to the following definition:
lim x->a f(x) = L if and only if lim x->a- f(x) = L and lim x-> a+ f(x) = L
Infinite Limits
- occurs when y values become infinitely large when x approaches a.
- do not regard infinity as a number!!!!
- the limit of f(x) as x approaches a is negative infinity when the values of x decrease without bound.
- the line x = a is a vertical asymptote of the curve y= f(x) if at least one of the following is true:
*can't insert infinity symbol, so I will represent using the word*
- lim x->a f(x) = infinity
- lim x->a- f(x) = infinity
- lim x->a+ f(x) = infinity
- lim x->a f(x) = negative infinity
- lim x->a- f(x) = negative infinity
- lim x->a+ f(x) = negative infinity
4 steps in determining the limit of a function:
- try plugging in values
- simplify algebraically
- look at a graph
- if all of the above don't work, then the limit D.N.E. (does not exist)
Sample Problem: (a simple, yet important concept)
(my internet lags so i apologize for not being able to post pictures)
what is the limit as x approaches 2 from the left? the right? does limx->2 exist? (refer to graph in link)
http://www.math.uncc.edu/~bjwichno/spring2005_math1241_002/Review_Calc_I/Images/1_over_x_2.gif
answer: no, because as x gets arbitrarily close to 2, the values become infinitely small (as x approaches from left) and infinitely large (as x approaches from right). there is, however, an asymptote at x=2.
links:
your ol' classic wiki favorite:
http://en.wikipedia.org/wiki/Limit_of_a_function
and this one, which most likely sums it up better than i do!http://www.math.hmc.edu/calculus/tutorials/limits/
and THIS ONE, is a cool simulator graph for limits. it's quite addicting... o.O http://www.math.psu.edu/courses/maserick/limit/limit.html
I hope that helped...and I believe it is the Lauren's turn?
Ending with a quote today...but I've always liked this quote from the "Father of Modern Mathematics" (pretty sure we'll deal with his concepts later on)...
"I think, therefore I am..." - Rene Descartes.
well, that's it. jaa ne...mata ne! (cya, and until next time!)
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