Evaluate the limit? $lim_(x rarr 0) x^2sin(1/x)$

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Answer 1 · 2017-05-14T17:08:44

$lim_(x rarr 0) x^2sin(1/x) = 0$

We want to find:

$L = lim_(x rarr 0) x^2sin(1/x)$

graph{x^2sin(1/x) [-0.3268, 0.3302, -0.1632, 0.1654]}

Graphically, it looks as though $L=0$ so let us see if we can prove this analytical:

Let $z = 1/x$ then as $x rarr 0 => z rarr oo$

So then, the limit can be written:

$L = lim_(z rarr oo) (1/z)^2sinz$
$\ \ = lim_(z rarr oo) (sinz)/z^2$
$\ \ = 0$

As $|sin(z)| le 1$ and $1/z^2 rarr 0$ as $z rarr oo$

Evaluate the limit? lim_(x rarr 0) x^2sin(1/x) lim_(x rarr 0) x^2sin(1/x)