Assgt no 3
Ex1 Let 𝑓(𝑥) =
𝑥 2 −2𝑥
𝑥 2 −4
- 𝐹𝑖𝑛𝑑 lim 𝑓(𝑥) by substituting values of x “close to 2.”
𝑥→2
-Estimate the lim 𝑓(𝑥) by using a graph
𝑥→2
Ex2 For each of the following graphs, estimate the limit lim 𝑓(𝑥)
𝑥→2
Ex 3
-State the 𝜀 −δ definition of lim 𝑓(𝑥) = 𝑙
𝑥→𝑎
-State the Sequence-definition of lim 𝑓(𝑥) = 𝑙
𝑥→𝑎
-State the 𝜀 −δ -definition of continuity of 𝑓 at 𝑎
-State the Sequence-definition of continuity of 𝑓 at 𝑎
-State the limit-definition of continuity of 𝑓 at 𝑎
-Define discontinuity of 𝑓(𝑥) at 𝑥 = 𝑎
- Define continuity of 𝑓(𝑥) in an open interval ]𝑎, 𝑏[
- Define continuity of 𝑓(𝑥) in a closed interval [𝑎, 𝑏]
- Define continuity of 𝑓(𝑥) in ]𝑎, 𝑏]
- Define continuity of 𝑓(𝑥) in [𝑎, 𝑏[
- State and define the Types of Discontinuity of 𝑓 𝑎𝑡 𝑎
- Define:
lim 𝑓(𝑥) = 𝐿
lim 𝑓(𝑥) = 𝐿
𝑥→−∞
𝑥→+∞
lim 𝑓(𝑥) = + ∞
lim 𝑓(𝑥) = − ∞
𝑥→𝑎
𝑥→𝑎
lim 𝑓(𝑥) = +∞
𝑥→+∞
lim 𝑓(𝑥) = +∞
𝑥→−∞
lim 𝑓(𝑥) = −∞
𝑥→+∞
lim 𝑓(𝑥) = −∞
𝑥→−∞
Ex 4
- Compute lim (𝑠𝑖𝑛𝑥 + 𝑥 2 )
𝑥→𝜋
1
- Compute lim ( 𝑥→0 𝑥
1
𝑥 2 +𝑥
)
- Use the definition of the limit to prove that lim(5𝑥 − 4) = 6
𝑥→2
- Use the definition of the limit to prove that lim
1
𝑥→0 𝑥 2
= +∞
Ex 5
The relationship between left limit, right limit and limit
lim 𝑓(𝑥) = 𝐿 iff lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 𝐿
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
a) Determine in each of the following cases lim− 𝑓(𝑥), lim+ 𝑓(𝑥), lim 𝑓(𝑥)
𝑥→2
𝑥→2
𝑥→2
b) Determine in each of the following cases lim 𝑓(𝑥)
𝑥→0
𝑓(𝑥) ={
0 𝑥 < 0
1
𝑥≥0
1
𝑓(𝑥) ={ 𝑥
1
𝑥>0
𝑥≤0
Ex6 Let 𝑓 : [−1, 1] → R defined by 𝑓(𝑥) ={
-Compute : lim+ 𝑓(𝑥),
𝑥→0
lim 𝑓(𝑥) and,
0
1
𝑖𝑓 − 1 ≤ 𝑥 ≤ 0
𝑖𝑓 0 < 𝑥 ≤ 1
lim 𝑓(𝑥), lim 𝑓(𝑥), lim + 𝑓(𝑥), lim− 𝑓(𝑥),
𝑥→0−
𝑥→0
𝑥→1
𝑥→−1
lim 𝑓(𝑥)
𝑥→−1−
𝑥→1+
Ex7 Compute
lim
𝑥 3 −𝑥
𝑥→−∞ 3𝑥 3 −4𝑥+5
,
lim
𝑥+3𝑥 2
𝑥→+∞ 𝑥 3 +5
𝑥
𝑥+√𝑥+3
Notice: lim
=
lim
𝑥→ ∞ 5𝑥+6√𝑥+7 𝑥→∞ 5𝑥
,
lim
3√𝑥 2
𝑥→+∞ 𝑥+5
but lim
,
𝑥 2 +1
𝑥→1 3𝑥 3 −4𝑥+5
lim
√𝑥 2
𝑥→−∞ 𝑥+5
≠ lim
𝑥2
𝒙→𝟏 3𝑥 3
Ex8
- Determine if the function 𝑓(𝑥) = {
3𝑥 − 5 𝑖𝑓 𝑥 ≠ 1
is continuous at 𝑥 = 1
2 𝑖𝑓 𝑥 = 1
.
𝑥 2 + 2𝑥
𝑖𝑓 𝑥 ≤ −2
- Determine if the function 𝑓(𝑥) = {
is continuous
2
𝑖𝑓 𝑥 > −2
at 𝑥 = −2
𝑥−6
𝑖𝑓 𝑥 < 0
𝑥−3
- Determine if the function 𝑓(𝑥)={
2
is
𝑖𝑓 𝑥 = 0
√4 + 𝑥 2
𝑖𝑓 𝑥 > 0
continuous at x = 0 .
- Let 𝑓(𝑥) = 3𝑥 2 + 𝑥 − 1. Show that f is continuous at x = 1.
- Determine if the function ℎ(𝑥) =
𝑥 2 +1
𝑥 3 +1
𝑥 3 −27
𝑖𝑓 𝑥 ≠ 3
2
- Check the function 𝑓(𝑥) = { 𝑥 9−9
2
is continuous at 𝑥 = −1
𝑖𝑓 𝑥 = 3
for continuity at 𝑥 =
3 𝑎𝑛𝑑 𝑥 = −3
Ex9
- For what values of 𝑥 is the function 𝑓(𝑥) =
𝑥 2 +3𝑥+5
𝑥 2 +3𝑥−4
continuous
- For what values of x is the function 𝑓(𝑥) = √𝑥 2 − 2𝑥 is continuous
(First describe function f using functional composition)
𝑥−1
- For what values of x is the function 𝑓(𝑥) = 𝑙𝑛 (
) is continuous
𝑥+2
- Suppose we have the function 𝑓(𝑥) =
𝑥 2 −2𝑥+2
𝑥 4 +1
using a) sequence- definition /
b) limit properties show that 𝑓 is continuous on its domain.
- Continuity of 𝑓(𝑥) = √𝑥 at 𝑥 = −1,
𝑥=0
Ex10
Classify the following discontinuities:
1
1
𝑥
𝑥2
𝑓(𝑥) = , 𝑓(𝑥) =
𝑓(𝑥) = lnx , 𝑓(𝑥) =
, 𝑓(𝑥)=
𝑥+3
, 𝑓(𝑥)={
𝑥
7
|x|
𝑖𝑓 𝑥 ≠ 3
,
𝑖𝑓 𝑥 = 3
𝑥 2 −1
𝑥−1
, 𝑔(𝑥) = 𝑥 + 3 𝑖𝑓 𝑥 ≠ 0, Classify the discontinuity
at x = 3.
Ex11
- What are intervals of continuity of
1
1
𝑥
1
,
𝑥2
𝑓(𝑥)=√
,
𝑙𝑛𝑥,
𝑓(𝑥)= √𝑥 ,
𝑓(𝑥)= √−𝑥,
𝑓(𝑥) = {
𝑥
𝑥+1
𝑖𝑓 𝑥 < 0
𝑖𝑓 𝑥 ≥ 0
𝑥+3
𝑥−10
Ex12
a) State i)- Intermediate value theorem ii)- The Extreme Value Theorem
b) Show that 𝑝(𝑥) = 2𝑥 3 − 5𝑥 2 − 10𝑥 + 5 has a root somewhere in the interval
1, 2.
-Show that 𝑓(𝑥) = 𝑥 3 − 4𝑥 + 1 has a root in the interval [1, 2].
- Consider the function 𝑓(𝑥) =
𝑥 2 −2𝑥+2
1+𝑥 4
. Does there exist 𝑥 ∈ [0, 1] such that
3
(𝑥) = ?
4
𝑥2
-If possible, determine if 𝑓(𝑥) = 20 𝑠𝑖𝑛(𝑥 + 3) 𝑐𝑜𝑠( ) takes the following values
2
in the interval [0,5].
(a) Does (𝑥 10 ? (b) Does 𝑓𝑥  10 ?
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