(a) find the limit. lim x ? ( ? /2) + (b) find the limit lim x ? 0 (c) Find the

Question

(a) find the limit.

lim x ? (?/2)+

(b) find the limit  

lim x?0

(c) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it.

cos x 1 ? sin x find the limit. lim x ? (?/2)+ Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. lim x ? (?/2)+ Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x?0 Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x?0

Explanation / Answer

(a)
Using l'hospital rule
limit cos(x) / (1-sin(x)) = limit -sin(x)/cos(x) = -infinity

(b)
limit (e^(2x)-e^(-2x)-4x)/(x-sin(x))
= limit (2e^(2x)+2e^(-2x)-4)/(1-cos(x)) (hospital rule)
= limit (4e^(2x)-4e^(-2x)/sin(x)) (hospital rule)
= limit (8e^(2x)+8e^(-2x)/cos(x)) (hospital rule)
= 16

(c) it's a)

(d) limit x/atan(2x) = limit 1/(2/(1+4x^2)) (hospital) = limit (1+4x^2)/2 = 1/2


(e) limit (cos(x)-1+x^2/2)/(2x^4) = limit (-sin(x)+x)/(8x^3)=limit(-cos(x)+1)/24x^2=limit(sin(x))/48x=limit(cos(x))/48=1/48

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