How do you graph #f(x)=7/(x+4)# using holes, vertical and horizontal asymptotes, x and y intercepts?
Holes : None
Vertical asymptote :
Horizontal asymptote :
Holes are literally what they mean—a hole in a graph. It's when both numerator and denominator of the equation have an exact same factor. For example, in the equation
In our equation
Vertical asymptotes are vertical lines where the graph approaches and get closer to but NEVER touches.
To find whether there is/are vertical asymptote(s), we set the denominator of the function equal to
Our denominator of the function is
As said earlier, vertical asymptotes are vertical lines, meaning they start with
Horizontal asymptotes are horizontal lines where the graph approaches; however, it can be crossed.
The following are the rules for finding horizontal asymptotes:
In the following,
Let m be the degree of the numerator.
Let n be the degree of the denominator.

if m > n , then there is no horizontal asymptote

if m = n , then the horizontal asymptote is dividing the coefficients of the numerator and denominator

if m < n , then the horizontal asymptote is y=0
From the equation
Xintercepts are where the graph touches the xaxis. It's sort of similar to finding the vertical asymptote, but we look at the numerator instead.
We set the numerator equal to
This means that there are no xintercepts.
Yintercepts are the values of
As said, let's plug in
So there is an yintercept at
If you need more help or want to watch a video, feel free to watch this:
Sorry, I know it's kind of long!
Hope this helps!
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To graph the function f(x) = 7/(x+4), we can start by identifying the vertical and horizontal asymptotes, holes, xintercepts, and yintercept.
Vertical asymptote: The vertical asymptote occurs when the denominator of the function is equal to zero. In this case, x+4 = 0, which gives us x = 4. Therefore, the vertical asymptote is x = 4.
Horizontal asymptote: To find the horizontal asymptote, we need to examine the behavior of the function as x approaches positive or negative infinity. In this case, as x approaches positive or negative infinity, the function approaches 0. Hence, the horizontal asymptote is y = 0.
Hole: To determine if there is a hole in the graph, we need to check if there are any common factors between the numerator and denominator that can be canceled out. In this case, there are no common factors, so there is no hole in the graph.
Xintercept: To find the xintercept, we set y (or f(x)) equal to zero and solve for x. In this case, we have 7/(x+4) = 0. Since the numerator is never zero, there are no xintercepts.
Yintercept: To find the yintercept, we set x equal to zero and evaluate the function. In this case, we have f(0) = 7/(0+4) = 7/4. Therefore, the yintercept is (0, 7/4).
To summarize:
 Vertical asymptote: x = 4
 Horizontal asymptote: y = 0
 Hole: None
 Xintercept: None
 Yintercept: (0, 7/4)
Using this information, you can plot the graph of f(x) = 7/(x+4) accordingly.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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