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MAT137 Exams Questions and Answers A+ Graded, Exams of Applied Mathematics

Example 4.1.1 a) Solve the inequality |x − 1| − |x − 3| ≥ 5.

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1 Absolute Values & Inequations 绝对值 &不等式
解不等式主要都是我们初高中之前所学的内容。 MAT137 这门课程中,解不等式,尤其是解带
有绝对值的不等式将会是本章节极限证明的一个重要基础工具,所以我们首先来复习一些最为常
用的解绝对值不等式的技巧,同时,我们需要掌握并记忆一些非常重要的不等式的性质。
1.1 使用分类讨论方法解不等式
Example 4.1.1
a) Solve the inequality |x1|−|x3| 5.
b) Solve the inequality 1+ex
1ex<2for xR.
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1 Absolute Values & Inequations &

MAT

Example 4.1.

a) Solve the inequality |x − 1 | − |x − 3 | ≥ 5.

b) Solve the inequality 1+ 1 −eexx < 2 for x ∈ R.

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1.2 |x − a| < r

  • |x − a| < r
  • |x − a| ≤ r
  • 0 < |x − a| < r
  • r 1 < |x − a| < r 2 &

Example 4.2. Construct an inequality with solutions a, b ∈ R, 0 < a < b. a) x ∈ (a − b, a) ∪ (a, a + b)

b) x ∈ (a − b, a) ∩ (a, a + b)

c) x ∈ [a, b] \ {^ a+ 2 b^ }

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Example 4.4. Which intervals are wholly contained in the solution to |x + 1| < 1? a) [− 4 , 1] b) [− 2 , 0) c) [− 1 , 1] d) (− 1 , 0] e) None of the above Example 4.4. If |x + 3| < 1 , then: a) |x − 3 | < 7 b) |x − 3 | < 5 c) − 7 < x − 3 < − 5 d) |x − 3 | > 7 e) None of the above Example 4.4. What values of x will satisfy the inequality |x + 1| < 1? a) [1, 2] b) [− 3 , −2] c) (− 2 , 0) d) (− 5 , −3] e) None of the above

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Example 4.4. Determine whether the following statements are True or False: a) Given x < a, then x < b for all b ≤ a.

b) Given x < a ⇒ f (x) < b, then x < a ⇒ f (x) < c for c ≥ b.

c) Given x < a ⇒ f (x) < b, then x < c ⇒ f (x) < b for c ≤ a.

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  • Step 1: x approches a
  • Step 2: points move along the curve
  • Step 3: y approches L

notice

  • If f (x) approaches ±∞, we say the limit does not exist (DNE).
  • For the limit, we do not care what f (x) is exactly when x = a.

2.2 One-sided Limit & Two-sided Limit

One-sided Limit :

  • (^) xlim→a+ f (x) = L
  • (^) xlim→a− f (x) = L

Two-sided Limit :

  • lim x→a f (x) = L

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2.3 Uniqueness of Limits

Theorem: If lim x→a f (x) exists, it must be unique. In other words,

xlim→a f^ (x) =^ L^ if and only if lim x→a+^ f^ (x) =^ L^ =^ xlim→a−^ f^ (x). x → a f (x) a

two-sided limit DNE

  • one of one-sided limit DNE
  • one-sided limits are not equal

Graph :

  • Graph1 a a
  • Graph2 a a
  • Graph

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Example 2.4. Given the graph of function f (x). Compute the following limits or say they do not exist. Justify your answer.

(a) limx→ 0 f (f (x)) (b) limx→ 1 f (f (x))

(c) limx→ 2 f (f (x)) (d) limx→ 3 f (f (x))

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3 Precise Definition of Limits

Chapter MAT 1 2 3

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3.2 Case examples in limit proofs

Now, we want to prove whether the given constant is the value of the limit or not; that is

xlim→a f^ (x) =^ L^ or not.

  • Case1: Show lim x→a f (x) exists as given L. That is: given L, lim x→a f (x) = L.
  • Case2 Show lim x→a f (x) exists, but not as given L. That is: given L, lim x→a f (x) ̸= L
  • Case3: Show lim x→a f (x) DNE. That is: ∀L ∈ (^) R. (^) xlim→a f (x) ̸= L

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