Cho các hàm số f(x) = x^3 -1/2x^2 -3/2 và g(x) = x^2 -3x +1

* Hướng dẫn giải

Đáp án A.

$\begin{array}{l}f\text{'}\left(x\right)=3x^2-x;\\ f\text{'}\text{'}\left(x\right)=6x-1;\\ g\text{'}\left(x\right)=2x-3\end{array}$

$\begin{array}{l}\underset{x\to 0}{\lim }\frac{f\text{'}\text{'}\left(\sin 5x\right)+1}{g\text{'}\left(\sin 3x\right)+3}\\ =\underset{x\to 0}{\lim }\frac{6.\sin 5x-1+1}{2.\sin 3x-3+3}\\ =\underset{x\to 0}{\lim }\frac{3.\sin 5x}{\sin 3x}\end{array}$

$\begin{array}{l}=\underset{x\to 0}{\lim }\frac{\frac{3.\sin 5x}{5x}.5x}{\frac{\sin 3x}{3x}.3x}\\ =3.\frac{5.\underset{x\to 0}{\lim }\frac{\sin 5x}{5x}}{3.\underset{x\to 0}{\lim }\frac{\sin 3x}{3x}}\\ =3.\frac{5}{3}=5\end{array}$