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Summary of limits, part 11 the thing inside the limit [f (x + h) − f (x)]/h is not deﬁned at h = 0 but that doesn’t mean that it is usually hard to work out what the limit is; for example, when calculating the derivative of f (x) = x2 we have 6 November 2008 df x2 + 2xh + h2 − x2 = lim = lim (2x + h) = 2x (7) So the purpose of these notes is to bring all the diﬀerent limits together into one dx2 h→0 h h→0 summary; limits have a single idea behind them, but there is a set of diﬀerent deﬁnitions we have divided above and below by h, something that is possible provided h = 0; when, designed to deal with diﬀerent circumstances and this can get confusing. in fact, h = 0 is the point we are interested in; the fact we are working inside a limit means that this isn’t a problem. You can think of the limit as a piece of mathematical technology designed to get around The limit the fact that 0/0 isn’t deﬁned: it ﬁnds where the function was headed as it moves towards the place where it gives 0/0. Although it might be obvious what the value should be, that The idea is that if a function, f (x) gets closer and closer to some value L the closer x gets doesn’t mean we can ignore the need to work inside a limit, without an exact mathematical to some a then we say the limit of f (x) as x goes to a is L, or, language deﬁning what we mean by ‘going towards’, it would not be possible to prove Limit: (Informal deﬁnition). If the value of f (x) can be made as close as we like to L by theorems about diﬀerenciation. The precise mathematical language is taking values of x suﬃciently close to a but not equal to a then we write Limit: (Formal deﬁnition) lim f (x) = L (8) lim f (x) = L (1) x→a x→a if and only if for all ǫ > 0 there exists a δ > 0 such that for all |x − a| < δ |f (x) − L| < ǫ. The diﬃculty with the limit is that the examples sometimes seem a little contrived, So, in the formal deﬁnition we refered to ‘f (x) as close as we like to L’, this is made but, as we see, having the limit available as a piece of mathematical technology makes exact by saying that gor any ǫ, that is ∀ǫ > 0, we can make |f (x) − L| < ǫ. Before we said deﬁning diﬀerenciation much easier. The typical example for the limit is a function like that we can make f (x) as close as we like to L by taking ‘x suﬃciently close to a’, here we have made this exact, by saying we can ﬁnd a δ, that is ∃δ > 0 such that if |x − a| < δ x2 − x then |f (x) − L| < ǫ. The role of the formal deﬁnition is to make the ideas ‘as close as we f (x) = (2) x−1 like’ and ‘suﬃciently close’ into mathematics that can be used for proving theorems, such since as the the nice theorem. In this course however, we don’t actually prove this theorem. x2 − x x(x − 1) The niceness theorem is the theorem that says the limit behaves nicely, it says that if = (3) limx→a f (x) = L1 and limx→a g(x) = L2 then x−1 x−1 the x − 1 above and below cancel provided x = 1. When x = 1 this function is not deﬁned, lim [f (x) + g(x)] = L1 + L2 x→a it is zero over zero. Hence lim [f (x) − g(x)] = L1 − L2 x→a x−1 x=1 lim [f (x)g(x)] = L1 L2 (9) f (x) = (4) x→a undeﬁned x = 1 and, provided L2 = 0 However, the limit at x = 1 does not depend on the value at x = 1, so f (x) L1 lim = (10) x→a g(x) L2 2 2 x −x x − x x(x − 1) If L1 > 0 lim = lim lim x = 1 (5) x→1 x−1 x→1 x − 1 x − 1 x→1 lim f (x) = L1 (11) x→a Now, although this looks like a somewhat contrived example, it is not in fact so unusual, In fact, once we know that continuity is, this last result is a special case of a more general in deﬁnition of the derivative theorem: if lim x → af (x) = L1 and g(x) is continuous at x = L1 then df f (x + h) − f (x) lim g(f (x)) = g(L1 ) (12) = lim (6) x→a dx h→0 h √ √ 1 Conor Houghton, houghton@maths.tcd.ie, see also http://www.maths.tcd.ie/~houghton/231 So, if g(x) = x then this reduces to limx→a f (x) = L1 . 1 2 The one-sided limit we have illustrated above, if x < 4 then the closer x gets to 4, the closer the function gets to 4, however, if x > 4 then the closer x gets to 4, the closer f (x) gets to 3. Hence There are lots of reason why a limit might not exist: one obvious one is that there is a discontinuity. Consider lim f (x) = 4 x→4− x x<4 lim f (x) = 3 (17) f (x) = (13) x→4+ 7−x x>4 However, there is no limit as x → 4, just specifying that x is close to 4 doesn’t tell you that f (x) is close to a particular value, precisely because f (x) could be close to 4 or 3 depending on whether x is bigger or smaller than 4. 6 The formal deﬁnition is very similar to the formal deﬁnition for the limit, now, for the 5 limit from below we specify that x < a and, for the limit from above, that x > a. One-sided Limit: (Formal deﬁnition) 4 lim f (x) = L (18) x→a+ 3 if and only if for all ǫ > 0 there exists a δ > 0 such that for all x > a such that x − a < δ 2 |f (x) − L| < ǫ and lim f (x) = L (19) x→a− 1 if and only if for all ǫ > 0 there exists a δ > 0 such that for all x < a such that a − x < δ |f (x) − L| < ǫ. 1 2 3 4 5 6 7 Inﬁnite limits Another reason a limit, or even a one-sided limit might not exist is that the function There is a discontinuity at x = 4 and so there the limit approaches an inﬁnitely large negative or positive number, that is, at x = a the function is undeﬁned because it has the form 1/0. Obvious examples would be f (x) = 1/x lim f (x) (14) x→4 does not exist. It is nonetheless useful to have a mathematical language to describe this situation, that language is the one-sided limit. 4 One-sided Limit: (Informal deﬁnition). If the value of f (x) can be made as close as we like to L by taking values of x suﬃciently close to a and greater than a then we write 2 lim f (x) = L (15) x→a+ If the value of f (x) can be made as close as we like to L by taking values of x suﬃciently 0 -4 -2 0 2 4 close to a and less than a then we write lim f (x) = L (16) -2 x→a− Thus, for the limit from below, x → a−, we are only consider where the function is going as x approaches a from below, conversely, for the limit from above, x → a+, we are -4 only considering where the function is going as x approaches a from above. In the example 3 4 and f (x) = 1/x2 1 lim = ∞ (24) x→0+ x 10 The formal deﬁnition has to ﬁnd a mathematical language for the English ‘as large as we like’: Inﬁnite Limit: (Formal deﬁnition). 8 lim f (x) = ∞ (25) x→a 6 if and only if for all M > 0 there exists a δ > 0 such that for all x such that |x − a| < δ f (x) > M It does that by demanding that for all values of M, f (x) can be made bigger than M 4 by taking x close enough to a. There are similar deﬁnitions for the one-sided limits, so, for example, example 2 Inﬁnite one-sided limit from above: (Formal deﬁnition). lim f (x) = ∞ (26) x→a+ 0 -4 -2 0 2 4 if and only if for all M > 0 there exists a δ > 0 such that for all x > a such that x − a < δ According to our previous deﬁnitions, these functions have no limit at x = 0, there is f (x) > M no number they approach as x approaches zero. However, it is useful to deal with these examples by extending the deﬁnition of the limit to include inﬁnite limits. Inﬁnite Limit: (Informal deﬁnition). If the value of f (x) can be made as large as we like by taking values of x suﬃciently close to a but not equal to a then we write lim f (x) = ∞ (20) x→a if the value can be made as large a negative number as we like by taking values of x suﬃciently close to a but not equal to it, we write lim f (x) = ∞ (21) x→a There are also one-sided versions of this, for example If the value of f (x) can be made as large as we like by taking values of x suﬃciently close to a and greater than a then we write lim f (x) = ∞ (22) x→a+ Now, returning to f (x) = 1/x2 , the closer x gets to zero, the larger 1/x2 become, so 1 lim =∞ (23) x→0 x2 In contrast, 1/x is negative if x is negative and positive as x is positive, but the denominator gets smaller and smaller as x gets smaller, while the numerator does not, so 1 lim = −∞ x→0− x 5 6

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posted: | 11/27/2011 |

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