Re: Continuity limit graph help

Re: Continuity limit graph help

Quote:

Originally Posted by

**fkf** For the second question we need to know that

lim{x->a} f(x) = L <=> lim{x->a+} f(x) = lim{x->a-} f(x) = L

to consider the function having a limit L at x = a.

Thus we have no limit att x = 0 and neither at x = 1.

Why is there no limit at x = 1?

Re: Continuity limit graph help

Quote:

Originally Posted by

**emakarov** Why is there no limit at x = 1?

Plot a graph and then take the given limits from both sides and se that they're not the same.

lim{x->1+}f(x) =/= lim{x->1-}f(x)

We can say that we have two different onesided limits there. But usually when talking about "limit" then left+rightlimit needs to be the same.

Re: Continuity limit graph help

Quote:

Originally Posted by

**fkf** Plot a graph

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Re: Continuity limit graph help

Re: Continuity limit graph help

Look, are we speaking different dialects of English and that's why we don't understand each other? Are you talking about the limits of f' or of f itself? Aren't both one-sided limits equal to 1? On the Desmos graph you can even click the plot and move the point along the graph, and it will show you the coordinates.

In post #15 you wrote that we have a removable discontinuity at x = 1. But this means (Wikipedia) that both one-sided limits are equal. Indeed 1/x = 1 when x = 1: this is the left limit. On the right of x = 1, f(x) = 1, so the right limit is also 1.