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## Precalculus

### Course: Precalculus>Unit 10

Lesson 14: Connecting limits at infinity and horizontal asymptotes

# Functions with same limit at infinity

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)
A limit at infinity (like any other limit) describes the behavior of a function but it isn't unique to that function. Many different functions can have the same limit at infinity.

## Want to join the conversation?

• Why is this important to understand?
• From the author:The important thing is to understand limits at infinity. We hope that this video helps understanding limits at infinity a little better.

We see in this video how limits at infinity say something about the behavior of the function as the x-values increase to infinity, but they don't say exactly how the function behaves. If the limit at infinity is 3, all we know is that the function approaches 3. We don't know how the function approaches 3. In fact, it can approach 3 in many different ways, as we can see in Sal's multiple examples.
• If the limit of x-->infinity is 3, then does that mean that the functions will never actually reach 3?
• Yes that is correct. What the statement means is that as x gets bigger and bigger, y will get closer and closer to 3. At infinity, y will equal 3. So obviously this means that the function will never actually reach 3. Hope this helps!
• Isn't it true that graph of Sin(x)/x is oscillating around asymptote upto infinity?
If wee look at Desmos, graph is again crosses asymptote after 200.
Ref : https://www.desmos.com/calculator/sr7dge3tyu
• Yes, but since each oscillation has a smaller magnitude than the one before, the function is still approaching the asymptote.
• Why isn't the purple function oscillating around y=3 when Sal is zoomed in at 200?
(1 vote)
• Think about it. The purple function is 1/x*sin(x) + 3. As x approaches infinity, 1/x becomes extremely close to 0. Since sin(x) is the only oscillating part, if 1/x*sin(x) becomes about 0, so does the oscillating. If you don't understand why sin(x) oscillates, I encourage you to watch the videos about it on Khan Academy.
• If a graph has both a vertical and horizontal asymptote and the question is asking us to find the limit of the graph as x approaches infinity, which one do we use to determine the answer?
(1 vote)
• Be careful though, unlike vertical asymptotes, a function can cross its horizontal asymptotes at certain points in its domain, and function might not approach the horizontal asymptote as x approaches infinity, it really depends on the function.
(1 vote)
• I don't get how the oscillating function gets closer and closer to 3. Doesn't the function crossing 3 mean it is equal to 3 at certain points?
(1 vote)
• Yes, it does. But that doesn't mean the limit at infinity isn't also 3.

The limit being 3 means that, for any error bound ε, there is an x-value (call it M) beyond which f(x) stays within ε of 3.

So if we want the function to remain within 0.1 of 3, we just need to find a point after which the amplitude of the sine function is less than 0.1. This is always the case whenever x≥10. Because when x≥10, 1/x≤0.1, and the function is
[something ≤0.1]·[something between -1 and 1]+3. That is, the function is within 0.1 of 3.
(1 vote)
• how did he find the limit at 3?
(1 vote)
• Find the y value where x approaches 3 from the left and right. If the left limit and right limit equal, the limit exists. However, if they are not equal, the limit does not exist.
Hope this helps! If you have any questions or need help, please ask! :)
(1 vote)
• evaluate:
3x^2-x-2/5x^2+4x+1 when limit approches infinity