These vertical asymptotes occur when the function's denominator, n (x), is zero (not the numerator). Near the values x = 1 and x = -1, the graph practically vertically ascends or descends, and the function tends to + or -. With the equation x = 1, the graph shows a vertical asymptote. Because the value of the function is not defined there, this point must be avoided in any calculation involving this function.

A vertical asymptote can also occur if the function is constant. For example, f(x) = k has **a vertical asymptote** when k is itself a vertical asymptote. A linear function, such as y = mx **\+ c**, has **two vertical asymptotes**: x = -m/c and x = m/c. A quadratic function, such as y = x^2, has only one vertical asymptote at x = 0. A cubic function, such as y = x^3, has none.

A function can have more than one vertical asymptote. For example, the function f(x) = x^3 - 9x + 24 has asymptotes at both x = 3 and x = -3 because x^3 becomes very large when x gets close to these values.

- What happens at a vertical asymptote?
- How do you find the vertical asymptote of a parabola?
- When looking for vertical asymptotes, Why do we set the denominator equal to zero?
- Why can there be an infinite number of vertical asymptotes?
- Are vertical asymptotes limited?
- How do you find the vertical asymptote of a trig function?
- What is the horizontal asymptote of this graph?

Vertical asymptotes may be obtained by solving the equation n(x) = 0, where n(x) is the function's denominator note: this only applies if the numerator t(x) for **the same x value** is not zero. Determine the function's asymptotes. Because the denominator of this parabola is x - 4, its vertical asymptote will be found by solving x - 4 = 0 which gives us x = 9. Therefore, the parabola has one vertical asymptote at x = 9.

A horizontal asymptote can be found by solving t(x) = 0 for x. In this case, the equation becomes n(x) = x - 4 = 0 which implies that the parabola has no horizontal asymptotes.

To determine if there are any other types of asymptotes, we need to know what kind of parabola it is. We can tell this from the sign of **the leading coefficient**. If the leading coefficient is positive, then the curve is said to be upwardly concave; if it's negative, then the curve is downwardly concave. A parabola is said to have vertical asymptotes if and only if the leading coefficient is zero or negative. A parabola has **horizontal asymptotes** if and only if the leading coefficient is zero or positive.

When a vertical line is an asymptote, the graph approaches **the vertical line**. Because the graph turns vertical as it approaches the line, the vertical line is a representation of what the graph looks like as it approaches the line. We locate the vertical asymptotes by setting the function's DENOMINATOR to zero.

The vertical asymptote is a point at which the function is undefined and the function's limit does not exist. This is because, as one approaches the asymptote, even minor changes in the x-value create arbitrarily enormous variations in the function's value. For example, if we take the sin function and approach its vertical asymptote which is also equal to **-sin(180 degrees**) = cos(90 degrees) because sin(180 degrees) = 1, even small changes in x will result in sin(x) being near but not exactly equal to -1.

Because the function is undefined at the asymptote, it has no limit as x approaches the asymptote from the right. Because of this, the function can always reach negative values by approaching the asymptote from the right. Similarly, it can also reach positive values by approaching the asymptote from the left.

Here are some examples of functions with **vertical asymptotes**: f(x) = x^3, g(x) = e^x, h(x) = sin x.

For example, if we take the log of **very large numbers**, we find that the logarithm of 2 is 0 while the logarithm of 10^300 is 3. We can see that there is no way to define what the logarithm of a billion is; it might be anything.

As another example, consider the function f(x) = 1/x. There are two obvious answers when we ask how high the function goes: (1) Infinity or (2) -Infinity. But there is no way to specify either of these answers without using some kind of limit. In fact, if we try to calculate a limit for this function, we find that it doesn't exist. The reason is simple: as x gets big, so does f(x). If we change our definition of what it means to "get big", then we can make f(x) do anything we want it to do. For example, if we change our definition of "big" to mean "almost equal to 1", then f(x) will be close to 1 for sufficiently large values of x.

Vertical asymptotes do not exist. To locate the vertical asymptotes, we set the function's denominator to zero and solve. In **this case**, the vertical asymptotes are at ±180 degrees.

Horizontal asymptotes are lines that the graph approaches. Identifying **horizontal asymptotes** If the denominator's degree (the biggest exponent) is larger than the numerator's degree, the horizontal asymptote is the x-axis (y = 0). Otherwise, the horizontal asymptote is the value of the denominator y = log10(x).

In this problem we are given that the function is invertible, so it has an inverse. The inverse of a function is another name for a quantile function. There are several ways to find the quantile function for a distribution. One way is to use the inverse cumulative distribution function (ICDF), which in this case is the inverse hyperbolic cosine function. The ICDF for a distribution with mean equal to 50 and standard deviation equal to 10 is shown below:

The inverse hyperbolic cosine function is not defined for values less than or equal to 0. This occurs when the argument is a fraction, such as 0.5. To fix this issue, we need to restrict the domain of the function to (0, 1). We can do this by changing the condition to if arg <= 0 then return NaN.

Now that the function is restricted to (0, 1), we can find the quantiles by evaluating the inverse hyperbolic cosine at different values of x.