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Final

MATH 1200 Study Guide - Final Guide: Asymptote, Product Rule, Constant FunctionPremium

4 pages121 viewsFall 2016

Department
Mathematics
Course Code
MATH 1200
Professor
Matthew Demers
Study Guide
Final

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MATH 1200 8 Dec 2016
LIMITS
Tangent Lines
A tangent line is a line that locally touches a function at only one point. To find the slope of a tangent line, at a
given point a:
m= lim
h0
f(a+h)f(a)
h
Anormal line is the line perpendicular to the tangent; they have a slope of 1/m.
Limit Rules
Assuming that limxaf(x)and limxag(x)both exist, where a, c R.
1. limxa[f(x)±g(x)] = limxaf(x)±limxag(x)
2. limxacf(x) = c·limxaf(x)
3. limxaf(x)g(x) = limxaf(x)·limxag(x)
4. limxa
f(x)
g(x)=limxaf(x)
limxag(x)
5. limxah(f(x)) = h(limxaf(x)), for h(x)continuous.
Factor Theorem
If f(p) = 0 for any polynomial f(x),(xp)is a factor of f(x).
Ex. Find limx2
2x4+ 17x3+ 31x262x168
6x5+ 5x4153x3+ 8x2+ 660x400.
If you directly substitute x= 2 into the above function, you get 0
0. This means that (x2) is a factor for
both polynomials in the function. Factor (x2) out of each polynomial, using synthetic division, then try direct
substitution again.
One-sided Limits
Functions can sometimes be discontinuous at certain points, this is especially true for piece-wise functions. A limit
exists only when a function is continuous, this is determined by evaluating a limit approaching xfrom both the left
and right.
limxaf(x) = Lmeans the limit of f(x)approaching xfrom the left.
limxa+f(x) = L+means the limit of f(x)approaching xfrom the right.
If L̸=L+, the limit doesn’t exist.
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