Advanced Calculus And Analysis by I. Craw

By I. Craw

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Sketch for interest — not part of the course). Pick lim x→∞ f (x) −l < g (x) > 0 and choose a such that for all x > a. Then pick K such that if x > K, then g(x) − g(a) = 0. By Cauchy, f (c) f (x) − f (a) = g (c) g(x) − g(a) for all x > K. Note that although c depends on x, we always have c > a. 1 f (x) − f (a) f (x) g(x) − g(a) . 1 as x → ∞. = (Rates of growth) One interest in these results is to see how fast functions grow as x → ∞. This is explored further in the exercises. But important results are: • The function ex increases faster than any power of x.

0, there is some δ > 0 Note that we exclude the possibility that x = a when we consider a limit; we are only interested in the behaviour of f near a, but not at a. In fact this is very similar to the definition we used for sequences. Our main interest in this definition is that we can now describe continuity accurately. 4. Definition. Say that f is continuous at a if limx→a f (x) = f (a). Equivalently, f is continuous at a iff given > 0, there is some δ > 0 such that whenever |x − a| < δ, then |f (x) − f (a)| < .

The lines y = x and y = −x are also plotted. 3 One sided limits Although sometimes we get results directly, it is usually helpful to have a larger range of techniques. 4. 9. Definition. Say that limx→a− f (x) = l, or that f has a limit from the left iff given > 0, there is some δ > 0 such that whenever a − δ < x < a, then |f (x) − f (a)| < . 10. Example. Define f (x) as follows:  3 − x if x < 2; f (x) = 2 if x = 2;  x/2 if x > 2. Calculate the left and right hand limits of f (x) at 2. 4. RESULTS GIVING CONINUITY 35 Solution.

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