Unformatted text preview: : near 0 and to the left of 0 : x = 0 − δx
left
sin(0 − δ x)
− δx
sin(− δ x)
}={
}=1
} ={
f(x) = {
(0 − δ x)
− δx
− δx
Note how we simplify ffirst and then take the limit. We can simplify first because
irst
take
simplify ir
δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step.
here
division by zer
ero
step
2. LIMIT step : L imit {1} = 1 δx → 0 f(x) = − 4π − 3π − 2π − 1π 0 +1π sin (x)
x +2π +3π +4π . . . x * Note: we may infer this from the basic definition of sin (θ) = opposite side / hypoteneuse.
When θ is infinitely small, say δ θ , then sin ( δ θ ) = r δ θ/r = δ θ . Another way is to look at the expansion
of sin(x). When x becomes infinitely small, say δ x , then only the first term in the expansion matters. 32 Limit from the RIGHT
1. TENDS TO step : near 0 and to the right of 0 : x = 0 + δx
right
δx
sin(0 + δ x)
sin(+ δ x)
} =1
}= {
}={
f(x) = {
δx
(0 + δ x)
+ δx
Note how we simplify ffirst and then take the limit. We can simplify first because
irst
take
simplify ir
δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step.
here
division by zer
ero
step
2. LIMIT step : L imit δx → 0 {1} = 1 sin(x)
imit
imit
Since, L →  { f(x) } = 1 = L→ + { f(x) } , we say L imit { x } = 1 .
x
0
x
0
x→0 Value { f(x) } = Value
x=0 x=0 sin(x)
{ x } = 0 0 , which is something undefined.
/ Value
Example 9 : We now present an example of a function f(x) where x = a f(x)
L imit
exists everywhere (i.e. a can be any real number), but the x → a f(x) does NOT
exist anywhere.
f(x) = +1 if x is rational
rational { 0 if x is irrational
irr
ir Recall what we said about the rationals and irrationals. Between any two rationals
there are infinitely many rationals. And between any two rationals there are also
infinitely many irrationals. So we can see that as x TENDS TO a, the function f(x) will
keep fluctuating. The function f(x) will be either 0 or 1. It will not take on a single
specific value. Hence we can say that the Limit f(x) does NOT exist. Since a can
x→a
be any point on the real number line, the L imit does NOT exist anywhere.
On fi...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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