Limits at infinity rules in details:
As shown in the first graph, function tends to positive infinite limits from either side of line x=c. So these function will have limits to infinity at x = c. In the second graph, function tends to positive infinite but from left side it tends to negative infinite from right side of line x=c, so limit to infinity is different in both the cases which means that limits for function are not valid or don't exist. In both the cases, vertical line x = c is called vertical asymptotes of function. In third case, function tends to have y=b value when x tends to positive infinity. This horizontal line y = b is called horizontal asymptotes of function.
For any trigonometric function, limits at infinity are as given below. If g(x)= sin(x) then g only varies between 1 to 1 no matters what value x takes. Let’s have a look at Limits at infinity rules.

If function’s left hand limit and right hand limit at any point exists, but not equal, then limit at that point does not exist.

If RHL & LHL of a function at any point say x0 exist and equal, then limit at that point exists and lim x > x0 g(x) = lim x > a + g(x) = lim x > a  g(x) is true.
Process for finding limits at infinity :
We studied about the limits to infinity rules and infinite limit. Now we will be going through the infinite limits at infinity. If for g(x), g tends to positive infinite value in accordance with x tending to some value x0, then lim x>x0 g(x)=+∞. Similarly if g tends to negative infinite value in accordance with x tending to some value x0, then lim x>x0 g(x)=∞. The graph given above depicts this picture clearly. It's time for us to learn about finding limits at infinity through examples. So here are some of the limits at infinity examples for you.
(1) Solve infinite limits for
2) Solve infinite limits for
The answer to this function comes infinite and hence it has no horizontal asymptotes.
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