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18 3 EXAMPLES OF WHAT IS A LIMIT? { 1 x ≥ 0 (c) f(x) = −1 x < 0 at x = 0 Now will we just nd limit here without trying to see what is the RHL and LHL?? Here we have to nd the LHL and RHL separately and check if both are equal. If both are equal then the function has limits at x = 0 or else it doesnt have a limit. RHL = lim x→0 + f(x) = lim 1 x→0 + (since right of 0 f(x) = 1) = 1 LHL = lim x→0 − f(x) = lim (−1) x→0 − (since left of 0 f(x) = −1) = −1 And we see that LHL ≠ RHL Hence the limits doesnt exist in this case. 3.8 Practise Problems in Limits 1. lim x→0 + sin x |x| 2. lim θ→0 tan θ θ (Solution : 1) (Solution : 1) 3. Find the anwers to the following functions, given D f : [0, 3] and is represented in the adjoining gure (a) f(1)
19 (b) lim x→1 + (c) lim x→1 − (d) lim f(x) x→1 (e) lim f(x) x→2 + (f) lim x→2 − (g) f(2) Solutions- a:2,b:1,c:1,d:1,e:0,f:-1,g:0 Part III Lecture 3 4 Derivatives Denition. Derivative of a function y = f(x) at a point x = a is dened as dy f(x) − f(a) dx∣ = lim x→a x=a x − a = lim h→0 f(a + h) − f(a) h Above we found derivative of the function at some particular point x = a. What if we want to nd the derivative at any point x? We just replace a by x! Simple right? Notation : If given y = f(x), dy dx = d dx g(x) = g′ (x) So the derivative of a function y = f(x) at any point x is given as dy f(x + h) − f(x) dx∣ = lim at any x h→0 h Example. If f(x) = sin x, g(x) = x + 1 x − 1 then we have
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 and 14: 3.3 Rotating a marble tied to a cor
Page 15 and 16: 3.5 Denition of Limit 15 3.5 Deniti
Page 17: 3.7 Worked out problems in limits 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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