Calculus- Early Transcendentals, 2021a

80 Limits
Exercises for Section 3.2
Exercise 3.2.1 Give an ε–δ proof of the fact that lim
x→4
(2x − 5)=3.
Exercise 3.2.2 Let ε be a small positive real number. How close to 2 must we hold x in order to be sure
that 3x + 1 lies within ε units of 7?
3.3 Computing Limits: Graphically
In this section we look at an example to illustrate the concept of a limit graphically.
The graph of a function f (x) is shown below. We will analyze the behaviour of f (x) around x = −5,
x = −2, x = −1 andx = 0, and x = 4.
Observe that f (x) is indeed a function (it passes the vertical line test). We now analyze the function at
each point separately.
x = −5:Observethatatx = −5 there is no closed circle, thus f (−5) is undefined. From the graph we
see that as x gets closer and closer to −5 from the left, then f (x) approaches 2, so
lim
x→−5 − f (x)=2.
Similarly, as x gets closer and closer −5 from the right, then f (x) approaches −3, so
lim f (x)=−3.
x→−5 +
As the right-hand limit and left-hand limit are not equal at −5, we know that
lim f (x)
x→−5
does not exist.