110 4.4 L’Hôpital’s Rule (c) lim x→1 x−1 √x+1− √2 . (d) lim x→0 ex −e−x ln(1+x) . (e) lim x→1 lnx sin(πx) . 4.4.2 For the problems below use L’Hôpital’s rule as many times as appropriate to determine the limits. (a) lim x→0 1−cos2x xsinx . (b) lim x→0 (x−π 2) 2 1−sinx . (c) lim x→0 x−arctanx x3 . (d) lim x→0 x−sinx x−tanx . 4.4.3 Use the relevant version of L’Hôpital’s rule to compute each of the following limits. (a) lim x→∞ 3x2 +2x+7 4x2 −6x+1 . (b) lim x→0+ −lnx cotx . (c) lim x→∞ π 2 −arctanx ln(1+1 x) . (d) lim x→∞ √xe−x. (Hint: first rewrite as a quotient.) 4.4.4 Prove that the following functions are differentiable at 1 and -1. (a) f (x) = x2e−x2 , if |x| ≤1; 1 e , if |x| >1. (b) f (x) = arctanx, if |x| ≤1; π 4 signx+ x−1 2 , if |x| >1. 4.4.5 ▷Let P(x) be a polynomial. Prove that lim x→∞ P(x)e−x =0. 4.4.6 ▶Consider the function f (x) =(e− 1 x2 , if x̸ =0; 0, if x =0. Prove that f ∈Cn(R) for every n∈N.
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