let-f-x-x-x-2-sh-xt-sin-xt-dt-calculate-lim-x-0-f-x-

Question Number 74015 by mathmax by abdo last updated on 17/Nov/19

l e t f (x) = \int_{x}^{x^{2}} \frac{s h (x t)}{s i n (x t)} d t

c a l c u l a t e {l i m}_{x \to 0} f (x)

Commented by mathmax by abdo last updated on 18/Nov/19

f (x) = \int_{x}^{x^{2}} \frac{s h (x t)}{s i n (x t)} d t =_{x t = u} \frac{1}{x} \int_{x^{2}}^{x^{3}} \frac{s h (u)}{s i n (u)} d u

\exists c \in] x^{2}, x^{3} [/ \int_{x^{2}}^{x^{3}} \frac{s h (u)}{s i n (u)} d u = s h (c) \int_{x^{2}}^{x^{3}} \frac{d u}{s i n u}

\int_{x^{2}}^{x^{3}} \frac{d u}{s i n u} =_{t a n (\frac{u}{2}) = t} \int_{t a n (\frac{x^{2}}{2})}^{t a n (\frac{x^{3}}{2})} \frac{1}{\frac{2 t}{1 + t^{2}}} \times \frac{2 d t}{1 + t^{2}} = {[l n ∣ t ∣]}_{t a n (\frac{x^{2}}{2})}^{t a n (\frac{x 3}{2})}

= l n ∣ \frac{t a n (\frac{x^{3}}{2})}{t a n (\frac{x^{2}}{2})} ∣\sim l n ∣ x ∣ (x \to 0)

c = λ x^{2} + (1 - λ) x^{3} w i t h 0 < λ < 1 \Rightarrow f (x) \sim \frac{s h (c)}{x} l n ∣ x ∣

\sim s h (λ x^{2} + (1 - λ) x^{3}) l n ∣ x ∣ \sim (λ x^{2} + (1 - λ) x^{3}) l n ∣ x ∣

= x^{2} l n ∣ x ∣ (λ + (1 - λ) x) \to 0 (x \to 0) \Rightarrow {l i m}_{x \to 0} f (x) = 0