Lim x → a f x = L if f x is very close to L by...

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Review Chapter 1 Definition of Limits and Non-existence of Limits: (1) lim x a f ( x ) = L if f ( x ) is very close to L by taking x sufficiently close to a , but x 6 = a . (2) lim x a + f ( x ) = L ( lim x a - f ( x ) = L ) if f ( x ) is very close to L by taking x sufficiently close to a , and x > a ( x < a ). (3) lim x a f ( x ) Does Not Exist if (case i) lim x a + f ( x ) = ±∞ or lim x a - f ( x ) = ±∞ , (case ii) lim x a + f ( x ) 6 = lim x a - f ( x ), or (case iii) The graph of f oscillates infinitely many times as x approaches a . -3 -4 -1 1 2 3 1 2 3 4 -1 -2 -3 -4 5 4 -2 -5 Problem 1. From the above the graph of y = f ( x ), find the following limits: (a) lim x →- 5 f ( x ) (b) lim x →- 4 f ( x ) (c) lim x →- 3 f ( x ) (d) lim x →- 2 - f ( x ) (e) lim x →- 2 + f ( x ) (f) lim x →- 1 f ( x ) (g) lim x 0 f ( x ) (h) lim x 1 - f ( x ) (i) lim x 1 + f ( x ) (j) lim x 1 f ( x ) (k) lim x 2 f ( x ) (l) lim x 3 f ( x ) (m) lim x 5 - f ( x ) (n) lim x 5 + f ( x ) 1
How to solve the limit problem : lim x a f ( x ) g ( x ) (Case 1) lim x a g ( x ) 6 = 0 = lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) Problem 2. (a) lim x π/ 4 tan x + 1 sin x (b) lim x 1 x 2 - 4 x + 1 (Case 2) lim x a g ( x ) = 0 and lim x a f ( x ) 6 = 0, that is, c 0 form = lim x a f ( x ) g ( x ) DOES NOT EXIST . (Three possibilities : , -∞ , or DNE )

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