Limits (introduction) Alex Karassev

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 3.996001 0.003999 2.1 4.41 0.41 2.01 4.0401 0.0401 2.001 4.004001 0.004001

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?

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 2.997001 0.9999 2.99970001 x (x3-1)/(x-1) 1.1 3.31 1.01 3.0301 1.001 3.003001 1.0001 3.00030001 Guess: (x3 - 1) / (x - 1) approaches 3 as x approaches 1

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

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)

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

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 ≈ 3.14 / 180 = 0.0174444… Calculator gives: sin1° ≈ 0.017452406

Table of values for sinθ / θ 0.5 0.958851077 0.4 0.973545856 0.3 0.985067356 0.2 0.993346654 0.1 0.998334166 0.01 0.999983333 0.001 0.999999833 0.0001 0.999999998

Graph of y = sinθ / θ y 1 θ -π π

Danger of guessing What happens to when x approaches 0? x f(x) 0.01 0.010099828 0.001 0.0011 0.0001 0.0002 0.00001 0.00011 What is our guess?

Danger of guessing More values… x f(x) 0.001 0.0011 0.0001 0.0002 0.00001 0.00011 0.000001 0.000101 0.0000001 0.0001001 0.00000001 0.00010001 What is our guess now?

Solution As x approaches 0: sinx approaches sin0 = 0 cosx approaches cos0 = 1 So sinx + cosx / 10000 approaches 0 + 1 / 10000 = 0.0001

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!

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

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.)

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

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

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

Theorem

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

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)

Infinite limits: one-sided limits y x

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:

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

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)

Limit does not exist!