MA1S11 (Dotsenko) Solutions to Tutorial/Exercise Sheet 4
Week 5, Michaelmas 2013
1. Which of the following limits exist (as finite or infinite limits)? Explain your answer
and compute them.
3−5x3
3.
x→+∞ 1+4x+x
• lim
3−5x3
3.
1+4x+x
x→−∞
• lim
Solution. Clearly,
3−5x3
also lim
3
x→−∞ 1+4x+x
3−5x3
1+4x+x3
=
−5x3
x3
−5x3
= lim
x→−∞
x3
·
3
5x3
4
1+ 2 + 13
x
x
1−
3−5x3
3
1+4x+x
x→+∞
, so lim
−5x3
3
x→+∞ x
= lim
= −5 and
= −5.
2. Which of the following limits exist (as finite or infinite limits)? Explain your answer
and compute them.
√
x2 +x
.
3x−1
x→−∞
• lim
• lim
x→+∞
√
x+1−
√ x .
√
√
Solution. For the first limit, we note that
x2 +x
3x−1
=
x2
√
1+ x1
1
3x(1− 3x
, so
√
x2 + x
|x|
1
= lim
=− .
x→−∞ 3x − 1
x→−∞ 3x
3
√
√
√
2 −(√x)2
√
√
For the second limit, we note that x + 1 − x = ( x+1)
=
x+1+ x
lim
lim
√
x→+∞
x+1−
√
1 √
,
x+1+ x
so
√ x = 0.
3. Which of the following limits exist (as finite or infinite limits)? Explain your answer
and compute them.
• lim
1−cos x
.
sin x
• lim
tan x−sin x
.
x3
x→0
x→0
Solution. For the first limit, we have
1 − cos x
1 − cos x x2
1 − cos x
x
1
= lim
·
= lim
· lim
· lim x = · 1 · 0 = 0.
2
2
x→0
x→0 sin x x→0
x→0
sin x
x
sin x x→0
x
2
lim
1
For the second one, we have
tan x − sin x
= lim
x→0
x→0
x3
lim
sin x
cos x
− sin x
sin x(1 − cos x)
=
= lim
3
x→0
x
x3 cos x
sin x
1 − cos x
1
1
1
= lim
lim
=1· ·1= .
lim
2
x→0 x x→0
x→0 cos x
x
2
2
4. Show that
1
≤ |x|
x
= 0.
−|x| ≤ x sin
for all x 6= 0, and explain why lim x sin x1
x→0
Solution. Since | sin t| ≤ 1 for all t, we have |x sin x1 | = |x|| sin x1 | ≤ |x|, therefore
−|x| ≤ x sin x1 ≤ |x|. Since lim |x| = lim (−|x|) = 0, the Squeezing Theorem guaranx→0
x→0
tees that lim x sin x1 = 0.
x→0
(
sin x1 , x 6= 0,
Does there exist a choice of a for
5. Let us consider function f (x) =
a, x = 0.
which this function is continuous at x = 0? Explain your answer.
Solution.
No. If this were the case, this would mean that lim sin x1 = a. However,
x→0
this limit does not exist, since as x approaches zero, since sin x1 oscillates between −1
and 1 in every interval containing zero.
2