Exchanging limits

## Exchanging limits
            ---
            ## Exchanging limits
            #### Experiment
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0 0 auto;margin-right:2vw;">

Evaluate $\displaystyle \lim\_{x\to0^+} \left( \lim\_{y\to0^+} \left( x^y \right) \right)$

Evaluate $\displaystyle \lim\_{y\to0^+} \left( \lim\_{x\to0^+} \left( x^y \right) \right)$

</div>
            <div style="flex:1;margin-left:2vw">

</div>
            </div>
            
            ---
            ## Exchanging limits
            #### Experiment
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0 0 auto;margin-right:2vw;">

Evaluate $\displaystyle \lim\_{x\to0^+} \left( \lim\_{y\to0^+} \left( x^y \right) \right)$

Evaluate $\displaystyle \lim\_{y\to0^+} \left( \lim\_{x\to0^+} \left( x^y \right) \right)$

</div>
            <div style="flex:flex:1;margin-left:2vw">

$\displaystyle = \lim\_{x\to0^+} \left( x^0 \right) = \lim\_{x\to0^+} \left( 1 \right) = 1 \vphantom{\left(\lim\_{y\to0^+}\right)}$.

$\displaystyle = \lim\_{y\to0^+} \left( 0^y \right) = \lim\_{y\to0^+} \left( 0 \right) = 0 \vphantom{\left(\lim\_{x\to0^+}\right)}$.

</div>
            </div>

So there is danger in exchanging limits.
            
            ---
            ## Exchanging limits
            #### Experiment
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:0 0 auto;margin-right:2vw;">

Evaluate $\displaystyle \lim\_{x\to0^+} \left( \lim\_{y\to0^+} \left( x^y \right) \right)$

Evaluate $\displaystyle \lim\_{y\to0^+} \left( \lim\_{x\to0^+} \left( x^y \right) \right)$

</div>
            <div style="flex:flex:1;margin-left:2vw">

$\displaystyle = \lim\_{x\to0^+} \left( x^0 \right) = \lim\_{x\to0^+} \left( 1 \right) = 1 \vphantom{\left(\lim\_{y\to0^+}\right)}$.

$\displaystyle = \lim\_{y\to0^+} \left( 0^y \right) = \lim\_{y\to0^+} \left( 0 \right) = 0 \vphantom{\left(\lim\_{x\to0^+}\right)}$.

</div>
            </div>

So there is danger in exchanging limits.

To resolve the question
            > When can we exchange limits?

we need the concept of uniform continuity.