What does Almost always mean, and why is this result true? I've seen examples of when this works, of course, but why should it "almost always" work?
Show that $\lim_{x,y\to 0^+} x^y$ is "almost always" equal to $1$
2
$\begingroup$
real-analysis
-
0Aha! You snuck a $y$ in the limit. Now the question, and the phrase "almost always", makes more sense. – 2012-12-10
1 Answers
1
If $(x,y)$ approaches $(0,0)$ from within a sector in the first quadrant bounded by lines $y=ax$ and $y=bx$, then the limit of $x^y$ is $1$. You can show that by having it first approach along those two lines, and then squeezing. So it can approach a limit other than $1$ only if it follows a path having either the $x$-axis or the $y$-axis as a tangent line.