MATH 231 - Scientific Calculus I
Evaluating Limits for Functions Given by Formula
Recall that we have organized the function formulas for this course as falling into one of two categories:
•the Basic Three
•the Five Ways that functions are built from the Basic Three
With this organization in mind, it is easy to list out the properties of limits (known as the Limit Laws that
we will need to evaluate limits in this course. Here are the Limit Laws, for each of these categories:
the Basic Three
•Exponential Functions f(t)=bt,where bis a positive constant.
lim
t→cf(t)=f(c).
•Trig Functions f(t)=sin(t)andg(t)=cos(t).
lim
t→cf(t)=f(c) and lim
t→cg(t)=g(c)
•Power Functions f(t)=tp,where pis a constant.
lim
t→cf(t)=f(c), provided the domain of fcontains an open interval which includes c.
Functions built from the Basic Three
•Arithmetic Combinations: Example: Addition. (The others are similar, and are listed on page 108 of
our text.) If lim
t→cf(t) and lim
t→cg(t) both exist, then,
lim
t→c(f(t)+g(t)) = lim
t→cf(t) + lim
t→cg(t).
•Function Composition: h(t)=f(g(t)). If lim
t→cg(t)=Land lim
t→Lf(t)=Mboth exist, then
lim
t→ch(t)=M.
•Inverse Functions:
–Logarithms: f(t)=loga(t), where ais a positive constant. If c>0, then
lim
t→cf(t)=f(c) (i.e. lim
t→cloga(t)=loga(c)).
–Inverse Trig Functions: f(t)=sin
−1(t)org(t) = cos−1(t). If −1<t<1, then
lim
t→cf(t)=f(c) and lim
t→cg(t)=g(c)
•Piecewise Functions:
–If cis not a break point of the piecewise function f(t), treat it like any other function when
evaluating the limit lim
t→cf(t).
–If cis a break point of the function, evaluate the limit by working out the two one-sided limits,
and checking if they are equal.
•Domain Restriction: Compute the limit as with other functions, paying attention to allowable values
of the independent variable.
