Question

Найти асимптоты функции f(x)=x/2-1/x

Answer 1

Область определения функции: $D(f)=(-\infty;0)\cup(0;+\infty)$

Вертикальные асимптоты: $x=0$

$\displaystyle k= \lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{ \frac{x}{2} - \frac{1}{x} }{x} =0.5$

$\displaystyle b= \lim_{x \to \infty} f(x)-kx=0$

$y=kx+b=0.5x$ - наклонная асимптота

Горизонтальной асимптоты нет.

Answer 2

Finding the Asymptotes of the Function f(x) = x/2 - 1/x

To find the asymptotes of the function f(x) = x/2 - 1/x, we need to consider both vertical and horizontal asymptotes.

Vertical Asymptotes: Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a certain value. In this case, we need to find the values of x for which the denominator of the function becomes zero.

The denominator of the function is x, and it becomes zero when x = 0. Therefore, x = 0 is a vertical asymptote of the function.

Horizontal Asymptotes: Horizontal asymptotes occur when the function approaches a certain value as x approaches infinity or negative infinity. To find the horizontal asymptotes, we need to examine the behavior of the function as x approaches infinity and negative infinity.

As x approaches infinity, the term x/2 becomes dominant, and the function approaches positive infinity. Therefore, y = +∞ is a horizontal asymptote as x approaches infinity.

As x approaches negative infinity, the term x/2 becomes dominant, and the function approaches negative infinity. Therefore, y = -∞ is a horizontal asymptote as x approaches negative infinity.

To summarize: - Vertical asymptote: x = 0 - Horizontal asymptotes: y = +∞ (as x approaches infinity) and y = -∞ (as x approaches negative infinity)

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