Question

☁️

23🙏

limit is L. Take ε =
arrive at a
contradiction.]
23. If the function f is defined by
prove that limx→o f(x) does not exist.
24. By comparing Definitions 2, 3, and 4, prove T
lim f(x) = L____if and only if
lim_f(x) =
x→a
N
25. How close to -3 do we have to take x so that
1
10,000
(x + 3)4
27. Prove that lim
26. Prove, using Definition 6, that lim
F11
000
f(x) =
=
J 3
金
26
F12
28. Suppose that limx→a f(x) = ∞ and limx→a g(x)
c is a real number. Prove each statement.
届
(a) lim [f(x) + g(x)] = ∞
x-a
- 儿
鄉
in the definition of a l
5
x→−1− (x + 1)³
(1)
0 if x is rational
if x is irrational
Insert
+ 11
=
PrtSc
77%
33
x-a-
x→-3 (x + 3)4
-00元
Delete
Backspace
28°C
Fn
Q
Home
Num
Lock
End

Answer 1

我就是我

Assume lim x→0 f(x) = L exists

Consider epsilon=1/2, exist delta＞0
s.t. if 0＜|x-0|＜delta, then |f(x)-L|＜epsilon

let 0＜|x1-0|＜delta, and x1 is rational number;
0＜|x2-0|＜delta, and x2 is irrational number

We can find x1 and x2 for sure because rational numbers and irrational numbers are both dense

If 0＜|x1-0|＜delta, and 0＜|x2-0|＜delta
Then |f(x1)-L|＜epsilon and |f(x2)-L|＜epsilon
⇒ |0-L|＜1/2 and |1-L|＜1/2
⇒ -1/2＜L＜1/2 and 1/2＜L＜3/2 → ←
Thus, lim x→0 f(x) D.N.E.#

☁️

哈囉！如果直接寫如果lim x→0 f(x)要存在
lim x→0+ f(x)要=lim x→0- f(x)
但有理數無理數都是稠密的
所以無法確定 故也無法確定lim x→0 f(x)
這樣可以嗎？

☁️

謝謝泥！

我就是我

想法是對的，但是不夠嚴謹
圖中是另一題的正解，忽略開頭那句，後面就是這題的解答
和你的想法比較接近

我就是我

如果要抄的話
第一句和thus後面別抄
但是in fact後面要抄