如何找到垂直和水平渐近线？

函数是一种接受输入变量并提供结果的运算符。当一个量依赖于另一个量时，就会创建一个函数。函数的一个有趣特性是每个输入对应一个输出。换句话说，当且仅当它将集合 B 的每个元素恰好分配给集合 A 的一个元素时，两个集合（例如集合 A 和集合 B）之间的这种运算符称为函数。当所有输入和输出值都绘制在笛卡尔平面，称为函数图。

渐近线

这种与函数的整个图或图的一部分非常接近的假想线称为渐近线。在绘制函数图形时，渐近线非常有用，因为它们可以帮助您考虑曲线不应交叉的线。换句话说，渐近线是函数图收敛的点。在绘制函数时，我们很少需要绘制渐近线。

渐近线的类型

• 水平渐近线：水平渐近线是一条水平线，显示函数在图的极端边缘处的行为方式。但是，该函数很有可能会越过渐近线甚至触及它。对于具有多项式分子和分母的函数，存在水平渐近线。有理表达式是这些函数的名称。函数的水平形式为 y = k。

• 垂直渐近线：垂直渐近线是一条垂直线，它指向但不构成函数图形的一部分。该图永远不会穿过它，因为它发生在函数域之外的 x 值处。一个函数可能有多个垂直渐近线。

寻找水平渐近线

为了计算水平渐近线，考虑的点是给定函数的分子和分母的度数。确定函数水平渐近线的标准如下：

• 当分子和分母的度数相同时：将前导变量的系数相除，求水平渐近线。
• 如果分子的次数小于分母的次数：水平渐近线在y = 0 处，即x 轴。
• 如果分子的次数大于分母的次数：给定的有理函数没有水平渐近线。

寻找垂直渐近线

为了确定有理函数的垂直渐近线，需要遵循两个步骤。这些是：

步骤一：尽可能减少给定的有理函数，取出任何公因数，并通过因式分解简化分子和分母。

第二步：将分母设为零并求解 x。 x 的值是函数的垂直渐近线。

示例问题

问题 1. 求函数的水平和垂直渐近线：f(x) = $\frac{x^2-3x}{x+5}$ .

解决方案：

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 1

Since the degree of the numerator is greater than that of the denominator, the given function does not have any horizontal asymptote.

Vertical Asymptote:

Since the function is already in its simplest form, just equate the denominator to zero to ascertain the vertical asymptote(s).

⇒ x + 5 = 0

⇒ x = −5

问题 2. 二次函数可以有任何渐近线吗？

解决方案：

A quadratic function is a polynomial, so it cannot have any kinds of asymptotes. Since the polynomial functions are defined for all real values of x, it is not possible for a quadratic function to have any vertical asymptotes. Also, since the function tends to infinity as x does, there exists no horizontal asymptote either. 

问题 3. 求函数的水平和垂直渐近线：f(x) = $\frac{3x^2+6x}{x^2+x}$ .

解决方案：

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 2

Since the degree of the numerator is equal to that of the denominator, the horizontal asymptote is ascertained by dividing the leading coefficients.

⇒ HA = 2/2 = 1

Vertical Asymptote:

The function needs to be simplified first. $\frac{3x^2+6x}{x^2+x}=\frac{3x(x+2)}{x(x+1)}=\frac{3(x+2)}{x+1}$

Now that the function is in its simplest form, equate the denominator to zero in order to determine the vertical asymptote.

⇒ x + 1 = 0

⇒ x = −1

问题 4. 求函数的水平和垂直渐近线：f(x) = 10x ² + 6x + 8。

解决方案：

The given function is quadratic. A quadratic function is a polynomial, so it cannot have any kinds of asymptotes. Since the polynomial functions are defined for all real values of x, it is not possible for a quadratic function to have any vertical asymptotes. Also, since the function tends to infinity as x does, there exists no horizontal asymptote either. 

问题 5.求函数的水平渐近线：f(x) = 9x/x ² +2。

解决方案：

Degree of numerator = 1

Degree of denominator = 2

Since the degree of the numerator is smaller than that of the denominator, the horizontal asymptote is given by: y = 0.

问题 6. 求函数的水平和垂直渐近线：f(x) = x+1/3x-2。

解决方案：

Horizontal Asymptote:

Degree of the numerator = 1

Degree of the denominator = 1

Since the degree of the numerator is equal to that of the denominator, the horizontal asymptote is ascertained by dividing the leading coefficients.

⇒ HA = 1/3

Vertical Asymptote:

The function is in its simplest form, equate the denominator to zero in order to determine the vertical asymptote.

⇒ 3x – 2 = 0

⇒ x = 2/3

问题 7. 求函数的水平和垂直渐近线：f(x) = x ² +1/3x+2。

解决方案：

Horizontal Asymptote:

Degree of the numerator = 2

Degree of the denominator = 1

Since the degree of the numerator is greater than that of the denominator, the given function does not have any horizontal asymptote.

Vertical Asymptote:

Since the function is already in its simplest form, just equate the denominator to zero to ascertain the vertical asymtptote(s).

⇒ 3x + 2 = 0

⇒ x = −2/3