Math 101A, Fall 2025
Tutorial (Week 3)
Infinite Limits and the Squeeze Theorem
Another way to justify infinite limits is by explaining the behaviour
of the function using the language of limits.
1.
Determine the following limits. Note: The related function must be rewritten suitably so
that the reader can clearly see the behaviour of the function.
sin x
(a) lim− 2
x→4 x − 2x − 8
tan2 x sec x
(b) lim
π+
x→ 2
tan x
x→ 2 1 + sin x
(c) lim3π
Math 101A, Fall 2025
Tutorial (Week 3)
The goal of this problem is twofold: (i) to test your understanding
of the Squeeze Theorem, and (ii) to practice writing a complete
argument using the Squeeze Theorem, with a given inequality.
2.
For each of the following parts, state whether the given inequality (near x = 1) can be used
to determine lim f (x) using the Squeeze Theorem, and if yes, find the limit.
x→1
πx 2
≤ f (x) ≤
(a) tan
4
3−x
(b) 4x − 6 ≤ f (x) ≤ sec x
(c)
(x + 1)2 − 4
≤ f (x) ≤ 3 + cos(x − 1)
x−1
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Math 101A, Fall 2025
Tutorial (Week 3)
The last task is to learn how to construct simple inequalities.
3.
For each of the following limits, find an inequality and check that the Squeeze Theorem is
applicable. Be sure to state clearly where the inequality holds. The following facts
are helpful:
• −1 ≤ sin (f (x)) ≤ 1 whenever f (x) is defined.
• −1 ≤ cos (f (x)) ≤ 1 whenever f (x) is defined.
• |f (x)g(x)| = |f (x)| |g(x)|.
• |f (x)| ≤ C if and only if −C ≤ f (x) ≤ C.
π
(a) lim (x + 3)5sin( x+3 )
x→−3
2
(b) lim(t − 4) cos
t→2
1
t−2
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