By M. Batty
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If g(x) → 0 as x → ∞ then f (x) → 0 as x → ∞. • Exponential v. Polynomial: – If 0 < a < 1 and p is fixed then ax xp → 0 as x → ∞. – If b > 1 and p is fixed then bx xp → ∞ as x → ∞. • Logarithmic v. Polynomial: Let p < 0 be fixed. Then xp log(x) → 0 as x → ∞. 3 1. What is lim x→∞ sin(x) ? x2 Since for all x we have | sin(x)| 1, it follows that | sin(x) | x2 | x12 |. Since x12 → 0 as x → ∞, so does the sandwich theorems. sin(x) x2 by the second of 2. What is lim e−x x2010 ? x→∞ In the “exponential v.
F g : R → R by (f g)(x) = f (x)g(x), 4. f g : R → R by f g (x) = f (x) g(x) (if g(x) is never equal to 0), 5. kf : R → R by (kf )(x) = kf (x) (k ∈ R) and 6. f k : R → R by (f k )(x) = (f (x))k (where defined). Later, as we meet rules for manipulating limits, derivatives and integrals, you will see that they are usually concerned with building functions up in the ways above. 3 Inverse Functions A function f : R → R is injective if f (x1 ) = f (x2 ) always implies that x1 = x2 . If f (x) = 2x then f is injective because if 2x1 = 2x2 then x1 = x2 .
2. Use the above rules to differentiate x1999 + x2 + 1 . 2 The Chain Rule The chain rule tells us how to differentiate the composition of two functions. The Chain Rule: If f and g are functions from R to R, f is differentiable at x ∈ R and g is differentiable at f (x) then g o f is differentiable at x and (g ◦ f ) (x) = g (f (x))f (x) This theorem is also true for functions only defined on subsets of R, provided that we can compose them. Thus if f and g are differentiable (everywhere) then so is g ◦ f and we have (g ◦ f ) = (g ◦ f )f .
Essential Engineering Mathematics by M. Batty
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