Calculus 1 - Limits: Introduction to Limits, An Intuitive Understanding, Averages Rates of Changes and Secant Lines, Defining the Slope of A Curve, Instantaneous Rate of Change and Tangent Lines, Limits of Function Values, The Limit Laws, Eliminating Common Factors from Zero Denominators, Horner Me...
University Calculus ALL MODULUS (Chapters 1, 2, 3, 4) First Year University Calculus
FINAL EXAM WITH SOLUTION - MATH103 (Calculus 1) ALL CLASSES Calculus course covered
Class notes MTH 103 Calculus: Early Transcendentals Lecture 10
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CALCULUS 1: LIMITS
UI MATH 001
Study Notes & Practice Examples with Solutions
Calculus 1
LIMITS
,Limits
INTRODUCTION TO LIMITS
Calculus is the study of limits
What happens to the function f(x) as x gets close to some constant c ? Suppose that as an
object steadily ( moves forward we know its position at any given time. We denote the
position at time t by s (t). How fast is the object moving at time t = 1? We can use the formula
“distance equals rate times time” to find the speed (rate of change of position) over any
interval of time;
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
Speed =
𝑡𝑖𝑚𝑒
We call this the “average” speed over the interval since, no matter how small the interval is,
we never know whether the speed is constant over this interval. For example, over the
𝑠(2)−𝑠(1)
interval [1, 2], the average speed is ; over the interval [1, 1.2], the average speed
2−1
𝑠(1.2)−𝑠(1) 𝑠(1.02)−𝑠(1)
is ; over the interval [1, 1.02], the average speed is , etc. How fast
1.2−1 1.02−1
is the object traveling at time t = 1? To give meaning to this “instantaneous” velocity we
must talk about limit of the average speed over smaller and smaller intervals.
The idea of limit is essential to differential calculus. We will see that is necessary for finding
the slope of a tangent to the curve.
Consider the following table of values for f(x) = x2 in the vicinity of x = 2.
Notice that as x approaches 2 from the left, then f(x) approaches 4 from below.
Likewise, as x approach 2 from the right, then f(x) approach 4 from above.
We say that as x approach 2 from either direction, f(x) approach a limit of 4, and write
lim 𝑥 2 = 4
𝑥→2
, AN INTUITIVE UNDERSTANDING
If f(x) can be made as close as we like to some real number A by making x sufficiently
close to a, we say that f(x) approaches a limit of A as x approach a, and we write
lim 𝑓(𝑥) = A
𝑥→𝑎
We also say that as x approach a, f(x) converges to A.
Notice that we have not used the value of f(x) when x = a, i.e., f(a). This is very important
to the concept of limits.
5𝑥+ 𝑥 2
For example, if f(x) = and we wish to find the limit as x 0, it is tempting for us to
𝑥
0
simply substitute x = 0 into f(x). Not only do we get the meaningless value of , but also we
0
destroy the basic limit method.
= 5 + x if x 0
5𝑥+ 𝑥 2
Observe that if f(x) = then f(x) Is undefined if x = 0
𝑥
f(x) has the graph shown.
It is the straight line y = x + 5 with the point (0, 5) missing,
called a point of discontinuity of the function. However,
even though this point is missing, the limit of f(x) as x
approach 0 does exist. In particular, as x 0 from either
direction, f(x) 5.
5𝑥+ 𝑥 2
lim =5
𝑥→0 𝑥
Rather than graph functions each time to determine limits,
most can be found algebraically.
FIGURE 1
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