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Current time:0:00Total duration:2:21

Limits at infinity of quotients with trig (limit undefined)

Video transcript

let's see if we can figure out what the limit of x squared plus one over sine of X is as X approaches infinity so let's just think about what's going on in the numerator and then think about what's going on in the denominator so the numerator we have x squared plus one so as X gets larger and larger and larger as it approaches infinity well we're just squaring it here so it's going to assume Raydor is going to get even approaching infinity even faster so this thing is going to go to infinity as X approaches infinity now what's happening to the denominator here well sine of X we've seen this before sine of X and cosine of X are bounded they oscillate they oscillate between negative 1 and 1 so negative 1 is going to be less than or equal to sine of X which is going to be less than or equal to 1 so this denominator is going to oscillate so what does that tell us well we might be tempted to say well the numerator is unbounded goes to infinity and then the denominator is just oscillating between these values here so maybe the whole thing goes to infinity but we have to be careful because 1 this denominator is going between positive and negative values so this the numerator is just going to get more and more and more positive but we're being divided sometimes by positive values sometimes by a negative value so we're going to we're going to jump between positive and negative positive and negative and then you also have all these crazy asymptotes here every time X every time X every time sine of X becomes 0 well then you're going to be you're going to have a vertical asymptote this thing will not be defined so you have all these vertical asymptotes you're going to oscillate between positive and negative just larger and larger values and so this limit does not exist so does not exist does not exist and we can see that graphically we've described it in words just inspecting this expression but we can see it graphically if we actually look at a graph of this which I have right here and you can see that is X goes towards positive infinity as X goes to positive infinity we depending on which X we are we're kind of going we go we get really large then we had a vertical asymptote that we jump back down good really negative vertical asymptote up up down up down and just as the oscillations just get more and more extreme but we keep having these vertical asymptotes on a periodic basis so it's very clear that this limit does not exist