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1 as x oo. (We singled out null sequences as forming the simplest introduction to limits of sequences.

5. We have further If n is a positive integer, the function xn is continuous for all values of x; x-n is continuous except for x = 0. This can be proved directly (by proving that xi' — cn is small if x— c is small) or alternatively by induction, applying the theorem about the product of continuous functions to xn-1and x. We can now build up sums of multiples of powers of x to give Any polynomial is continuous for all values of x. 5 A quotient of two polynomials is continuous for all values of x for which the denominator is not zero.

State any values of x for which the following functions are discontinuous ,/{(x—a)/(b—x)), 11A1(x2 +1), tan x, sec x, 1/(1+ tan x). 2. Sketch the graph of the 'saw-tooth function' x—[x]-1. 3. Sketch the graphs of the functions [,lx], V[x], pointing out for what values of x there are discontinuities. 4. Prove that the function defined by f(x) = x sin (1/x) (x * 0), 1(0) = 0 is continuous for all values of x. Sketch its graph. 5. Prove that, if x2 — 6x+ 5 f(x) = x2 -9x+18' and k * 1, there are two values of x for which f(x) = k.

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