1 By Gökhan Bilhan Calculus 1 (Week 4)-Limits Part1 Denition(Limit) We write limx→c f (x) = L, if the value f (x) is close to the single real number L whenever x is close but NOT EQUAL to c. Remarks (1) x must approach to c from left and right. (2) c, need not to be in the domain. (3) x ̸= c, always. Example limx→0 |x| limx→0 |x| x 2 By Gökhan Bilhan Denition (One Sided Limits) We write limx→c− f (x) = K and say, limit from left (or left hand limit) if x approaches to c from left. In that case x < c. We write limx→c+ f (x) = K and say, limit from right (or right hand limit) if x approaches to c from right. In that case x > c. Theorem(Existence of a Limit) limx→c f (x) = A if and only if limx→c + f (x) = limx→c− f (x) = A At open holes, function is not dened. At closed (dotted) holes function is dened. limx→−1− f (x) =? limx→−1+ f (x) =? limx→−1 f (x) =? limx→1− f (x) =? limx→1+ f (x) =? limx→1 f (x) =? limx→2− f (x) =? limx→2+ f (x) =? limx→2 f (x) =? 3 By Gökhan Bilhan Theorem (Properties of Limits) Let f and g be two functions and assume that limx→c f (x) = L and limx→c f (x) = M where L and M are real numbers (i.e. both limits exist). Then 1-) limx→c k = k for any constant k . 2-) limx→c x = c. 3-) limx→c (f (x) + g(x)) = limx→c f (x) + limx→c g(x) = L + M . 4-) limx→c (f (x) − g(x)) = limx→c f (x) − limx→c g(x) = L − M . 5-) limx→c (f (x).g(x)) = limx→c f (x).limx→c g(x) = L.M . 6-) limx→c kf (x) = klimx→c f (x) = kL. f (x) limx→c f (x) L = = . g(x) lim M √ x→c g(x) √ √ 8-) limx→c n f (x) = n limx→c f (x) = n L note that L > 0 if n is even. 7-) limx→c Examples 1-) limx→3 (x2 − 4x) =? √ 2-) limx→−1 2x2 + 3 =? √ limx→−1 2x3 + 1 =? 4 By Gökhan Bilhan { x2 + 1, if 3-) f (x) = x − 1, if (a) limx→2− f (x) =? (c) limx→2 f (x) =? x<2 x>2 (b) limx→2+ f (x) =? (d) f (2) =? Exercises 1. Answer the following by looking at the following graph a-) limx7→4− f (x) b-) limx7→4+ f (x) c-) limx7→4 f (x) d-) f (4) =? e-) Is it possible to dene f (4) so that limx7→4 f (x) = f (4) f-) limx7→2− f (x) g-) limx7→2+ f (x) h-) limx7→2 f (x) k-) f (2) =? l-) Is it possible to dene f (2) so that limx7→4 f (x) = f (2) 5 By Gökhan Bilhan 2. Same Question 3. Let f (x) = 3x2 + 2x − 1 , nd x2 + 3x + 2 a) limx→−3 f (x) =? c) limx→2 f (x) =? b) limx→−1 f (x) =? 6 By Gökhan Bilhan x2 + x − 6 4. Let f (x) = , nd x+3 a) limx→−3 f (x) =? b) limx→0 f (x) =? c) limx→2 f (x) =? { x2 , if 5. Let f (x) = x − 1, if a) limx→1+ f (x) =? c) limx→1 f (x) =? x<1 x>1 b) limx→1−1 f (x) =? d) f (1). 7 By Gökhan Bilhan { 1 + mx, if 6. Let f (x) = 4 − mx, if x≤1 where m is a constant. x>1 a) Graph f for m = 1. b) limx→1− f (x) =? c) limx→1+ f (x) =? d) Find m so that limx→1− f (x) = limx→1+ f (x) 7. limh→0 [3(a + h) − 2] − (3a − 2) =? h 8 By Gökhan Bilhan Denition (Indeterminate Form) If limx→c f (x) = 0 and limx→c g(x) = 0,then limx→c to be indeterminate or more specically a Example limx→2 x2 − 1 =? x−1 0 indeterminate form. 0 f (x) is said g(x) 9 By Gökhan Bilhan Example limx→−1 x|x + 1| =? x+1 Example limx→−1,00000000001 x|x + 1| =? x+1 10 By Gökhan Bilhan Example limx→1,000001 Example If f (x) = Example limx→−1 √ |x| =? x x, then nd limh→0 f (2 + h) − f (2) =? h x−1 =?(Can we solve it yet?) x+1 11 By Gökhan Bilhan Exercises √ √ 1. limh→0 a+h− h 2. limx→a x2 − a2 =? x−a 3. limx→−2 a =? 2x2 + 3x − 2 =? x+2 1 1 − 4. limx→2 x 2 =? x−2 12 By Gökhan Bilhan √ 5. limx→−2 6. limx→2 x2 =? x x2 =? x x x 7. limx→0 =? √ limx→0 limx→0 limx→2 x2 =? x x2 =? x x =? x 13 By Gökhan Bilhan (Week 4)-Limits(Exercises) Exercises 1. limx→3 A) 23 2. limx→2 A)-1 x3 − 3x2 =? x2 − 3 B) 12 C)0 D)3 E)6 x3 − 8x + 8 =? x4 − 4x B) −1 7 C)0 D)1 E)2 √ 3− a−8 exists. 3. What is "a" to say the limit limx→2 x−2 A)12 B)11 4. limx→1− ( A)1 5. limy→x A)0 C)5 D)3 E)2 |1 − x| + x) =? 1−x B)2 C)−1 D)−2 E)0 y 3 − x3 =? y 2 − x2 3 B) x 2 C)2x 2 D) x 3 E)2 14 By Gökhan Bilhan 1 8 =? 6. limx→ 1 1 2 2 x − 4 x3 − A)− 43 B)− 34 C) 43 √ 3 x−4 7. limx→64 √ =? x−8 A)0 B) 13 D) 18 E) 21 1 Hint:( x 6 = t ) C) 23 D) 32 E)3 |x| , if x ̸= 0 8. Let f (x) = and x 3, if x=0 if limx→0+ f (x) = a and limx→0− f (x) = b , then what is a − b? A)−2 B)−1 C)0 D)1 E)2 15 By Gökhan Bilhan 2 if x , 9. Let f (x) = 3, if x + a, if x<3 x=3 x>3 For the above function f , what is a to say the limit at x = 3 exists? A)4 B)6 C)7 10. Look at the graphs D)8 E)9 By Gökhan Bilhan Answers: 1-) C , 2-) C , 3-)B , 4-)B , 5-)B , 6-)C , 7-)B , 8-)E , 9-)B 16
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