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Example 4.1.1 a) Solve the inequality |x − 1| − |x − 3| ≥ 5.
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Example 4.1.
a) Solve the inequality |x − 1 | − |x − 3 | ≥ 5.
b) Solve the inequality 1+ 1 −eexx < 2 for x ∈ R.
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^ }
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
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.
notice
One-sided Limit :
Two-sided Limit :
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
Graph :
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))
Chapter MAT 1 2 3
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.
