Crap, what an embarrassing mistake to make... Regardless, the picture is still largely meaningless. Does 81/3 = √81 x 3 really merit the title of "******* math"?
Actually the two sided limit is undefined. But as x approaches 8 from the left, the limit is -infinity. As x approaches 8 from the right, the limit goes to +infinity.
Sources: I'm Chinese. I'm born with this knowledge imbued in me.
The question is asking what the equation approaches as x gets closer to the limit but not what it is AT the limit. so in both it would be +- infinity because you are dividing over a number that is infinitely small in + or - depending on what direction you approach from. [url deleted]
is a graph of both of these questions (not really but it shows the idea)
No. Math has notation just like English has grammar. The limit doesnt exist because coming from the right and coming from the left the limits are different, meaning the overall limit does not exist.
Pic related. It's the same way with math. If it was limx->8- [1/(x-8)] then it would be true
This is why schools should just use standard limit notation. (parentheses) for approaches without bound, and [brackets] for reaches. so this would be (-∞,5) (5,∞) because the function never has an X value of negitive infinity, 5, or positive infinity.
Actually that's only true if you're assuming x=8, so y(x-8) would be infinity x zero which equals 1. What's more, in the case that (x-8) is zero or even less than one, y would have to be greater than one for the whole thing to equal 1.
I remember seeing the picture as a kid. At that time I was looking at it for 30 min trying to figure out how it is done. In the end I have come to a conclusion that I just can t do it.
On this very day in exactly 1 h I have a partial exam in limes and demand and supply functions. Is it irony that I see this picture today ?
In other words, the limit doesn't exist in R, which is the space that is generally assumed if none is given. The equation could still hold if the limit were taken in a different space, but one would generally expect something like that to be specified.
I think you're confusing upper and lower bounds/continuity with limit theory... the function is discontinuous and unbounded in the neighbourhood of x = 8 but the limit can still be equated to infinity... this just proves discontinuity.
Wrong.
Limit x->8+ (from the right) 1/(x-8) = infinity.
Limit x->8- (from the left) 1/(x-8) = -infinity
If limit x->a+ =/= limit x->a- then limit x->a Does not exist.