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14 3 EXAMPLES OF WHAT IS A LIMIT? But x can get closer and closer to 1 from two directions, one from left of 1 and other from right of 1. • x → 1 + As x approches to 1 from right of 1. what value does f(x) takes? x 1.1 1.01 1.001 1.0001 1.00001 f(x) 2.1 2.01 2.001 2.0001 2.00001 Can you make out what is happening as x → 1 + then f(x) → 2 • x → 1 − As x approaches to 1 from left of 1. What values does f(x) takes? x 0.9 0.99 0.999 0.9999 0.99999 f(x) 1.9 1.99 1.999 1.9999 1.99999 So you can see that here to the limit for f(x) is 2. Hence we conclude mathematically that x 2 − 1 lim x→2 x − 1 = 2
3.5 Denition of Limit 15 3.5 Denition of Limit If y = f(x) is given and as x → a , f(x) → l then we say that the limit of f(x) is l Mathematically that is written as lim f(x) = l x→a Now let us put some thought on the denition of limit we say as x → a, f(x) → l, now x tending to a means x comes closer to a. Since a lies on Real line x will come closer to a from left of a and right of a. For example, x → 2 means x will take values 1.9, 1.99, 1.999, 1.9999, . . . from left of 2 and 2.1, 2.01, 2.001, 2.0001, 2.00001, . . .. from right of 2. So the above denition, lim f(x) exists x→a actually means Note : here the equality in the mathematical way of writing the limit is not the usual equality. It is just a representation that f(x) is getting closer to l as x goes closer to a. 3.6 Formulae of limits LHL = lim f(x) exists x→a− RHL = lim f(x) exists x→a + & LHL = RHL = l sin x 1. lim x→0 x = 1, lim tan x cos x = 1 & lim = 1 x→0 x→0 x 2. lim x→0 (1 + x) 1/x = e = lim x→∞ (1 + 1 x )x a x − 1 3. lim = log a , a > 0 x→0 x 3.7 Worked out problems in limits 1. lim x→2 x 2 − 4 x − 2 x 2 − 4 lim x→2 x − 2 = (x − 2)(x + 2) lim x→2 x − 2 = lim (x + 2) x→2 (since here x → 2 So x − 2 ≠ 0) = 4 Above method implicitly means this as x → 2 =⇒ x + 2 → 4 & x − 2 → (x + 2)(x − 2) 0 =⇒ → 4 (since x − 2 is a factor both numertor − x − 2 denominator)
Page 1 and 2: Foundation of Calculus July 15, 200
Page 3 and 4: CONTENTS 3 5.4 Methods of Solving I
Page 5 and 6: 1.2 Dene slope for two points 5 1.2
Page 7 and 8: 1.4 Slopes of line 7 1.4 Slopes of
Page 9 and 10: 2.4 Domain and range of a function
Page 11 and 12: 11 Part II Lecture 2 3 Examples of
Page 13: 3.3 Rotating a marble tied to a cor
Page 17 and 18: 3.7 Worked out problems in limits 1
Page 19 and 20: 19 (b) lim x→1 + (c) lim x→1
Page 21 and 22: 21 dy dx = lim f(x + h) − f(x) h
Page 23 and 24: 4.1 Derivatives of other functions
Page 25 and 26: 4.3 Rate of change 25 Obviously, if
Page 27 and 28: 4.6 Chain Rule and friends 27 Simil
Page 29 and 30: 4.8 Derivative of a composite funct
Page 31 and 32: 5.3 Denite Integration 31 (a) d dx
Page 33 and 34: 5.3 Denite Integration 33 from the
Page 35 and 36: 5.4 Methods of Solving Integrals 35
Page 37: REFERENCES 37 References [1] Thomas

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