Limits, Continuity, and Trigonometric Limits

1. q = charge dq
dt
= electrical current
2. s = distance ds
dt
= speed
dT = temperature =
dx
1
Lecture 2 18.01 Fall 2006
4. Sensitivity of measurements: An example is carried out on Problem Set 1. In GPS, radio
signals give us h up to a certain measurement error (See Fig. 2 and Fig. 3). The question is
?L how accurately can we measure L. To decide, we find . In other words, these variables are
?h
related to each other. We want to find how a change in one variable affects the other variable.
L
s h
satellite
you
Figure 2: The Global Positioning System Problem (GPS)
s h
L
Figure 3: On problem set 1, you will look at this simplified “flat earth” model
2
?
Lecture 2 18.01 Fall 2006
Limits and Continuity
Easy Limits
x2 + x 32 + 3 12
lim = = = 3
x?3 x + 1 3 + 1 4
With an easy limit, you can get a meaningful answer just by plugging in the limiting value.
Remember,
lim
?f = lim f(x0 + ?x) ? f(x0)
x?x0 ?x x?x0 ?x
is never an easy limit, because the denominator ?x = 0 is not allowed. (The limit x x0 is
computed under the implicit assumption that x =? x0.)
?
Continuity
We say f(x) is continuous at x0 when
lim f(x) = f(x0)