✔ 最佳答案
1) It is indeterminate form of 0/0. Hence applying the L'Hospital rule, the limit is equal to:
lim (x → 0) [d(ex + e-x - 2)/dx]/[d(x sin 4x)/dx]
= lim (x → 0) (ex - e-x)/(sin 4x + 4x cos 4x)
still indeterminate form of 0/0. Applying the rula again:
= lim (x → 0) (ex + e-x)/(4 cos 4x + 4 cos 4x - 16x sin 4x)
= 1/4
2) lim (x → ∞) 5x+3/(26x/2 √5)
= (53/√5) lim (x → ∞) 5x/(26x/2)
= (53/√5) lim (x → ∞) (25/26)x/2
= 0 since 25/26 < 1
3) lim (x → -∞) (3 + 2x)/(3x + 9)
= (3 + 0)/(0 + 9) since all 2x and 3x vanish as x tends to negative infinity.
= 1/3
4) lim (x → ∞) (3 + 2x)/(3x + 9) = lim (x → ∞) [31-x + (2/3)x]/(1 + 32-x)= 0 since all 31-x and (2/3)x vanish as x tends to positive infinity5) -1/x <= (sin mx)/x <= 1/xHence by squeezing principle:lim (x → ∞) (sin mx)/x = 0
2011-05-23 23:21:21 補充：
I think you are referring to Q1 only.
BTW, let me ask you if you have learnt:
lim (x → ∞) (1 + 1/x)^x = e
this limit ?
2011-05-24 17:08:21 補充：
1) By this approach:
e^x = 1 + x + x^2/2 + x^3/3 + ...
e^-x = 1 - x + x^2/2 - x^3/3 + ...
Hence:
e^x + e^-x - 2 = x^2 + x^4/12 + ...
= x^2 (1 + x^2/12 + ...)
(e^x + e^-x - 2)/(x sin 4x) = x (1 + x^2/12 + ...)/sin 4x
2011-05-24 17:08:25 補充：
Thus:
lim (x → 0) (e^x + e^-x - 2)/(x sin 4x) = lim (x → 0) x (1 + x^2/12 + ...)/sin 4x
= lim (x → 0) (1 + x^2/12 + ...) lim (x → 0) x/sin 4x
= (1/4) lim (x → 0) 4x/sin 4x
= 1/4
