A constant factor can be taken out of the limit lim x→akf (x) = klim x→af (x) Limit of a composite function lim x→af (g(x)) = f (lim x→ag(x)) Limit of a continuous function If the function f (x) is continuous at x = a, then lim x→af (x) = f (a)Answer lim x → ∞ f ( x) = ∞ Example 7 Use the graph below to understand lim x → ∞ x sin 2 ( 2 x) The oscillations of this function continue to increase in amplitude, but since the function always returns to the x axis, we can't say the limit is infinite Answer lim x → ∞ x sin 2Chapter4 RealAnalysis 281 51 Disprove the claim If lim x→a f(x) = L, then either lim x→a f(x) = L or lim x→a f(x)= − L 52 Iflim x→a f(x)= ∞ andlim x→a g(x)= ∞,thenlim x→a f g = ∞ 53 Iflim x→a f(x)= ∞ andlim x→a g(x)= L ∈ R∗,thenlim x→a f(x) g(x) = ∞ 54 Iflim x→a f(x)= L,thenlim x→a (f(x)−L)= 055 Disprovethetwoclaims (a) Iflim x→a f(x)= L
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