Question

помогите вычислить предел функции (xy+sinx)/(xy+siny) когда x и y стремятся к бесконечности.

Answer 1

$lim_{x->\infty;y->\infty}\frac{xy+sinx}{xy+siny}=\\\\lim_{x->\infty;y->\infty}\frac{1+\frac{sin x}{x}*\frac{1}{y}}{1+\frac{sin y}{y}*\frac{1}{x}}=\\\\\frac{1+1*0}{1+1*0}=1$

Answer 2

Calculating the Limit of a Function

To calculate the limit of the function \( \frac{xy+\sin x}{xy+\sin y} \) as \( x \) and \( y \) tend to infinity, we can use the concept of limits in calculus.

The limit of a function as \( x \) and \( y \) tend to infinity can be determined by examining the behavior of the function as \( x \) and \( y \) become very large. Let's proceed with the calculation.

The given function is \( \frac{xy+\sin x}{xy+\sin y} \).

To calculate the limit as \( x \) and \( y \) tend to infinity, we can use the concept of limits and consider the behavior of the function as \( x \) and \( y \) become very large.

Solution

Using the given function \( \frac{xy+\sin x}{xy+\sin y} \), as \( x \) and \( y \) tend to infinity, we can consider the dominant terms in the numerator and denominator.

The dominant terms in the numerator are \( xy \) and \( \sin x \), and the dominant terms in the denominator are also \( xy \) and \( \sin y \).

As \( x \) and \( y \) tend to infinity, the terms \( xy \) become dominant in both the numerator and denominator.

Therefore, the limit of the function \( \frac{xy+\sin x}{xy+\sin y} \) as \( x \) and \( y \) tend to infinity is:

\( \lim_{{x,y \to \infty}} \frac{xy+\sin x}{xy+\sin y} = 1 \)

This result is obtained by considering the dominant terms in the numerator and denominator as \( x \) and \( y \) tend to infinity.