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Unformatted text preview: hing undefined.
/0
x  a
x=a
x=a
n
n
The reader must have guessed by now that L imit { xx  a } = n a n−1 .
 a
→
x a deriv tiv
We shall see this very special limit in its proper context when we study the derivative
deri
of f(x) = xn .
29 Example 6 : What is the LIMIT of f(x) = + √x as x TENDS TO 0 ?
f(x) = +√ x (0, 0) x Limit from the LEFT
1. TENDS TO step : near 0 and to the left of 0 :
left x = 0 − δx f(x) = √ (0 − δx) = √ − δx
A negative real number under the radical sign is a complex number. We are dealing
with real numbers and real valued functions. Here √ − δx is not real valued and
so it is not defined. f(x) = + √x is not defined for x < 0.
not left
2. LIMIT step : The LIMIT from the left does not exist.
Limit from the RIGHT
1. TENDS TO step : near 0 and to the right of 0 :
right x = 0 + δx f(x) = √ (0 +δx) = √δx
2. LIMIT step : L imit δx → 0 {√δx} = 0 Since L imit − f(x) ≠ L imit + f(x) we say : L imit f(x) does NOT exist.
x→0 x→0 x→0 Value
x = 0 f(x) = f(0) = √0 = 0
30 Example 7: What is the LIMIT of f(x) = 1 x as x TENDS TO 0 ?
/
Limit from the LEFT
1. TENDS TO step : near 0 and to the left of 0 :
left x = 0 − δx f(x) = f(0 − δ x) = { 1 (0 − δ x) } = − 1 δ x . Note that δx ≠ 0 .
/
/
2. LIMIT step : L imit δx → 0 { − 1 δ x }= − 1 0 = −∞ , something undefined.
/
/ Limit from the RIGHT
1. TENDS TO step : near 0 and to the right of 0 :
right x = 0 + δx f(x) = f(0 + δ x) = { 1 (0 + δ x) } =+1 δ x . Note that δx ≠ 0 .
/
/
L
2. LIMIT step : δ ximit0 {+1 δ x }= =+1 0 = +∞ , something undefined.
/
/
→
The LIMIT from the left does not exist. And, the LIMIT from the right does not exist.
left
right
1 = ∞ , something undefined.
VALUE { f(x) } = f(0) = / 0
x=0
+∞
/
f(x) = 1 x
2
1 ½
−∞ 0 ½1
−1
−2 −∞
31 2 ... +∞ sin(x)
as x TENDS TO 0 ?
x
We shall use the trigonometric identity sin (A ± B) = sin A cos B ± cos A sin B.
Also, when x is very small, we may say* : sin (x) = x . Example 8: What is the LIMIT of f(x) = Limit from the LEFT
1. TENDS TO step...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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