Question 992139
{{{f(x)}}}{{{""=""}}}{{{system(matrix(2,3,

(1-cos(ax))/(x*sin(x)),if,x<>0,
1/2,if,x=0))}}}
Find the value of "a" such that the function is 
continuous at x=0<pre>
f(x) will be continuous at x=0 if and only if

{{{matrix(2,2,

lim,(1-cos(ax))/(x*sin(x)),
"x->0","")}}}{{{""=""}}}{{{1/2}}}

We use L'Hopital's rule on the left side:

{{{matrix(2,2,

lim,(a*sin(ax))/(x*cos(x)+sin(x)),
"x->0","")}}}{{{""=""}}}{{{1/2}}}

We use L'Hopital's rule again on the left side:

{{{matrix(2,2,

lim,(a^2*cos(ax))/(-x*sin(x)+cos(x)+cos(x)),
"x->0","")}}}{{{""=""}}}{{{1/2}}}

{{{(a^2*cos(a*0))/(-0*sin(0)+2cos(0))}}}{{{""=""}}}{{{1/2}}}

{{{(a^2*1)/(-0*0+2*1)}}}{{{""=""}}}{{{1/2}}}

{{{a^2/2}}}{{{""=""}}}{{{1/2}}}

{{{a^2}}}{{{""=""}}}{{{1}}}

{{{a}}}{{{""=""}}}{{{"" +- 1}}}

Edwin</pre>