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1(a) Show that

(1 +1/n)n = 2 +nXk=2(1 1n )(1 2n)···(1 k1n )/k!.(1)

(10 points)

(b) By (a), show that the sequence an = (1 +1/n)n , n = 1, 2, 3, · · · , is increasing.

(10 points)

(c) Show that

(1 +1n)n < 2 +nXk=21/k!.

(5 points)

(d) Use the mathematical induction to show that k! 2k1.

(5 points)

(e) Conclude by (a) through (d) that

2 < (1 +1/n)n < 3.

(5 points)

Note: The increasing sequence (1+

1

n

)n , n = 1, 2, 3, · · · , bounded from above, then attains a limit, which is the celebrated Euler number e.

1. (a) Defifine

f(x, y) =x2y2/x2 + xy + y2 .

Verify that the domain of f is R2 \ {(0, 0)}.

(5 points)

(b) Prove that

lim(x,y)(0,0) f(x, y) = 0.

(10 points)

1. Prove that the function

f(x, y) :=x3y /x6 + y2

has no limit at (0, 0).

(5 points)

1. (a) Give an example f : R −→ R and g : R −→ R such that f is continuous and g is not continuous while g f is continuous.(5 points)

(b) Use the  δ defifinition to show that the function

f : x 0 7−→ √x

is continuous at x = 10.

Hint:

f(x) f(a) = x − √a = x  a/√x +a.

(5 points)

1. Give an example of a continuous function f : R subset S of R such that f(S) is not closed in R. −→ R and a closed  (15 points)

Hint: A compact set in R is bounded and closed. f sends a compact set to a compact set, which is closed in R. Thus, the set S you construct can never be bounded to begin with. So, you may want to try S := N

6(a). Let f : [a, b] −→  R be continuous with the property f(a) = f(b).Prove that f is not  one-to-one. (Hint: Explore the intermediate value theorem.) (10 points)

(b) Prove that a one-to-one continuous function g : [a, b]−→ R is either strictly increasing, i.e., f(x) < f(y) for any x < y, or strictly decreasing, i.e., f(x) > f(y) for any x < y. (10 points) EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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