1. Limits (introduction)
Alex Karassev

2. The concept of a limit What happens to x2 when x approaches 2?
Look at table of values: x x2
|x2 - 4|
1.9
3.61
0.39
1.99
3.9601
0.0399
1.999
2.1
4.41
0.41
2.01
4.0401
0.0401
2.001

3. Direct substitution
Thus we see that x2 approaches 4, i.e. 22, as x approaches 2
So we can just substitute 2 in the function x2
Is it always the case?

4. Example: direct substitution is impossible
Look at (x3-1)/(x-1) as x approaches 1
We cannot sub. x=1, however, we can substitute the values of x near 1 x
(x3-1)/(x-1)
0.9
2.71
0.99
2.9701
0.999
0.9999 x
(x3-1)/(x-1)
1.1
3.31
1.01
3.0301
1.001
1.0001
Guess: (x3 - 1) / (x - 1) approaches 3 as x approaches 1

5. Measuring the earth…
How to decide whether a planet is flat or "round"?
Look at circles of the same radius on a sphere and on the plane and compare their circumferences

6. Circumferences r r a Cr Cr θ Depends on θ and hence on the circle! R
Euclidean plane
Sphere of radius R r r R Cr a Cr θ R
Cr = 2π a
a = R sinθ and r = R θ
So Cr / r = (2π a) / r = (2π R sinθ) / (R θ)
= 2π (sinθ / θ)
Depends on θ and hence on the circle!
Cr = 2π r
So Cr / r = 2π
is constant! (does not depend on circle)

7. What if θ is very small? r Cr
When θ is small the piece of sphere enclosed by our circle is "almost flat", so we should expect that
Cr / r is very close to 2π
However, Cr / r = 2π (sinθ / θ)
Therefore, when θ is close to 0, sinθ / θ is close to 1 ! Cr θ
θ is close to 0
Guess: sinθ / θ approaches 1 as θ approaches 0
Equivalently: sinθ ≈ θ when θ is near 0

8. Example: what is sin 1° ? sinθ ≈ θ when θ is near 0
1° = π / 180 radians, so θ = π / 180
We have: sin1° = sin (π / 180) ≈ π / 180 ≈ / 180 = …
Calculator gives: sin1° ≈

9. Table of values for sinθ / θ
0.5
0.4
0.3
0.2
0.1
0.01
0.001
0.0001

10. Graph of y = sinθ / θ y 1 θ -π π

11. Danger of guessing What happens to when x approaches 0? x f(x) 0.01
0.001
0.0011
0.0001
0.0002
What is our guess?

12. Danger of guessing More values… x f(x) 0.001 0.0011 0.0001 0.0002
What is our guess now?

13. Solution As x approaches 0: sinx approaches sin0 = 0
cosx approaches cos0 = 1
So sinx + cosx / approaches / =

14. Informal definition We write
if we can make the values of f(x) as close to L as we like by taking x to be sufficiently close to a but not equal to a
Note: the value of function f at a, i.e. f(a), does not affect the limit!

15. Examples When we compute limit as x approaches 2, x is not equal to 2y 2 4
y=f(x)
When we compute limit as x approaches 2, x is not equal to 2

16. Does limit always exist?1
y=h(x)
If x approaches 0 from the left then h(x) is constantly 0
If x approaches 0 from the right then h(x) is constantly 1
Thus when x is close to 0 h(x) can be arbitrarily close (in fact, even equal to) 0 or 1 and we cannot choose either of this two numbers for the value of the limit of h(x) as x approaches 0
So does not exists (D.N.E.)

17. One-sided limits Limit of f(x) as x approaches a from the left
Limit of f(x) as x approaches a from the right

18. From the left
We write
if we can make the values of f(x) as close to L as we like by taking x to be sufficiently close to a and less than a x y a
y = f(x) L
f(a) x

19. From the right
We write
if we can make the values of f(x) as close to L as we like by taking x to be sufficiently close to a and more than a x y a
y = f(x) L
f(a) x

20. Theorem

21. Infinite limits Let f(x) = 1/x2
If x is close 0 then1/x2 is a very large positive number
Therefore the finite limit of 1/x2 as x approaches 0 does not exist
However, we may say that 1/x2 approaches positive infinity as x approaches 0 x y M

22. Infinite limits
We write if we can make the values of f(x) bigger than any number M by taking x to be sufficiently close to a and not equal to a
(note: think of M as of large positive number)
We write if we can make the values of f(x) smaller than any number M by taking x to be sufficiently close to a and not equal to a
(note: think of M as of large negative number)

23. Infinite limits: one-sided limitsy x

24. Vertical asymptote Definition
we say that a line x=a is a vertical asymptote for the function f if at leas one of the following is true:

25. Example What happens when x approaches 0? x sin(π/x) 1/2
sin(π/1/2)=sin(2 π)=0
1/3
1/4
1/n
sin(π/1/n)=sin(πn)=0

26. However… x
sin(π/x)
2/5
sin(π/2/5)=sin(5/2)π= sin(π/2 + 2 π) = sin(π/2) = 1
2/9 1
2/13
2/17
2/(4n+1)

27. Limit does not exist!