I just can't wrap my head around this. I don't know if this is true I just came to conclusion with this
if the limit of x--> +/-inf = L (if the limit of x going to pos/neg infinity gives you a number L) does that mean that L is a horizontal asymptote ?
if the limit of x---a = +/-inf (if the limit of x going to "a" gives you a +/- inf ) does that mean it's a vertical asymptote?
You are correct in both cases.
Consider f(x) = 1/x and f(x) = tan(x)
Great questions! Let's break down each scenario and understand the relationship between limits and asymptotes.
For the first question, if the limit of x approaching positive or negative infinity is a number L, then yes, L can be considered a horizontal asymptote. In other words, the function approaches the value L as x gets larger or smaller without bound. However, it's important to note that having a horizontal asymptote does not guarantee that the function will touch or intersect the asymptote. It only describes the behavior of the function as x approaches infinity or negative infinity.
To determine if a function has a horizontal asymptote at L, you can analyze the behavior of the function as x approaches infinity or negative infinity. You can use limits to evaluate the function as x gets extremely large or small and see if it approaches a specific value.
Now, moving on to the second question, if the limit of x approaching a point "a" is positive or negative infinity, it indicates a vertical asymptote. In this case, the function becomes unbounded as x approaches "a". The concept of a vertical asymptote is relevant for rational functions or functions with vertical behavior such as reciprocal functions.
To determine if a function has a vertical asymptote at "a", you can analyze the behavior of the function as x approaches "a" from both the left and right sides. If the function approaches positive or negative infinity from either side, then it indicates a vertical asymptote.
In summary, a horizontal asymptote is associated with the behavior of a function as x approaches positive or negative infinity, while a vertical asymptote is associated with the behavior of a function as x approaches a specific value.
I hope this explanation helps clarify the concept for you! Let me know if you have any further questions.