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