Calculus- Early Transcendentals, 2021a

102 Limits
When solving problems using the Squeeze Theorem it is also helpful to have the following theorem.
Theorem 3.41: Monotone Limits
If f (x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x
approaches a, thenlim f (x) ≤ lim g(x).
x→a x→a
Exercises for 3.6
Exercise 3.6.1 Compute the following limits.
sin(5x)
(a) lim
x→0 x
sin(7x)
(b) lim
x→0 sin(2x)
cot(4x)
(c) lim
x→0 csc(3x)
tanx
(d) lim
x→0 x
(e)
sinx − cosx
lim
x→π/4 cos(2x)
Exercise 3.6.2 For all x ≥ 0, 4x − 9 ≤ f (x) ≤ x 2 − 4x + 7. Findlim
x→4
f (x).
Exercise 3.6.3 For all x, 2x ≤ g(x) ≤ x 4 − x 2 + 2. Findlim
x→1
g(x).
Exercise 3.6.4 Use the Squeeze Theorem to show that lim
x→0
x 4 cos(2/x)=0.
3x + sinx
Exercise 3.6.5 Find the value of lim x→∞ . Justify your steps carefully.
x + cosx
3.7 Continuity
The graph shown in Figure 3.3(a) below represents a continuous function. Geometrically, this is because
there are no jumps in the graphs. That is, if you pick a point on the graph and approach it from the left and
right, the values of the function approach the value of the function at that point. For example, we can see
that this is not true for function values near x = 1 on the graph in Figure 3.3(b) which is not continuous at
that location.