1-masala. Hosila ta'rifidan foydalanib $$ f^{\prime}(0) $$ topilsin (agar mavjud bo'lsa). $$ 1.1 f(x)=\left\{\begin{array}{l}t g\left(x^{3}+x^{2} \sin \frac{1}{x} ight), x
eq 0 \ 0, x=0 .\end{array} \quad 1.2 f(x)=\left\{\begin{array}{l}\sin \left(x \sin \frac{3}{x} ight), x
eq 0, \ 0, x=0\end{array} ight. ight. $$

Question

1-masala. Hosila ta'rifidan foydalanib $$ f^{\prime}(0) $$ topilsin (agar mavjud bo'lsa). $$ 1.1 f(x)=\left\{\begin{array}{l}t g\left(x^{3}+x^{2} \sin \frac{1}{x}ight), x 
eq 0 \ 0, x=0 .\end{array} \quad 1.2 f(x)=\left\{\begin{array}{l}\sin \left(x \sin \frac{3}{x}ight), x 
eq 0, \ 0, x=0\end{array}ight.ight. $$

UpStudy AI · Accepted Answer

**1.1 Masala**

Berilgan funksiya
$$
f(x)=
\begin{cases}
\tan\Bigl(x^3+x^2\sin\frac{1}{x}\Bigr), & x\neq0,\[1mm]
0, & x=0
\end{cases}
$$
hosila ta'rifidan foydalanamiz:
$$
f'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0}\frac{\tan\Bigl(x^3+x^2\sin\frac{1}{x}\Bigr)}{x}.
$$

Kichik $$ x $$ qiymatlari uchun:
$$
\tan y\sim y\quad \text{(ya'ni, } \tan y = y+o(y) \text{, } y\to0\text{)}.
$$
Shu bilan
$$
\tan\Bigl(x^3+x^2\sin\frac{1}{x}\Bigr)\sim x^3+x^2\sin\frac{1}{x}.
$$
Demak,
$$
f'(0)=\lim_{x\to 0}\frac{x^3+x^2\sin\frac{1}{x}}{x}=\lim_{x\to0}\left(x^2 + x\sin\frac{1}{x}\right).
$$

Tahlil qilamiz:
- $$\lim_{x\to0} x^2=0.$$
- $$\lim_{x\to0} x\sin\frac{1}{x}=0,$$ chunki $$\left|x\sin\frac{1}{x}\right|\leq |x|$$ va $$ \lim_{x\to0}|x|=0.$$

Natijada:
$$
f'(0)=0+0=0.
$$

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**1.2 Masala**

Berilgan funksiya
$$
f(x)=
\begin{cases}
\sin\Bigl(x\sin\frac{3}{x}\Bigr), & x\neq0,\[1mm]
0, & x=0.
\end{cases}
$$
Hosila ta'rifi bo'yicha
$$
f'(0)=\lim_{x\to 0}\frac{\sin\Bigl(x\sin\frac{3}{x}\Bigr)-0}{x}=\lim_{x\to0}\frac{\sin\Bigl(x\sin\frac{3}{x}\Bigr)}{x}.
$$

Kichik $$ x $$ qiymatlari uchun $$\sin uchi atrofida chiziqli ya'ni:
$$
\sin\Bigl(x\sin\frac{3}{x}\Bigr)\sim x\sin\frac{3}{x}.
$$
Shu bilan
$$
f'(0)=\lim_{x\to 0}\frac{x\sin\frac{3}{x}}{x}=\lim_{x\to 0}\sin\frac{3}{x}.
$$

Ammo $$\lim_{x\to 0}\sin\frac{3}{x}$$ mavjud emas, chunki $$\frac{3}{x}$$ $$x\to0$$ holatda cheksizlikka qarab tebranadi va $$\sin$$ funktsiyasi osil tebranishi sababli cheksiz oscillatsiya qiladi.

Shunday qilib, $$ f'(0) $$ mavjud emas.

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**Xulosa:**

- **1.1 masala:** $$ f'(0)=0 $$.
- **1.2 masala:** $$ f'(0) $$ mavjud emas.
**1.1 Masala:** $$ f'(0) = 0 $$. **1.2 Masala:** $$ f'(0) $$ mavjud emas.

1-masala. Hosila ta’rifidan foydalanib topilsin (agar
mavjud bo’lsa).

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1-masala. Hosila ta’rifidan foydalanib topilsin (agar mavjud bo’lsa).