If f(x)-->L as x-->a and f(x)-->M as x-->a, then prove that L=M for 'f' be a function from R to R.
Please tell me its Solution with step by step...!
Thanks a lot for this.

Use the definition of limit:
"
if and only if for any [math]\epsilon>0 [math] there exists [math] \delta>0[\math] such that implies "

Then if , you can write . So you have from hypothesis
"
which translates to

for any [math]\epsilon>0 [math] there exists [math] \delta>0[\math] such that implies "

and

for any [math]\epsilon>0 [math] there exists [math] \delta'>0[\math] such that implies ".

Then choose very small, say : in this case you should have the corresponding ; put , and choose an arbitrary such that : you have now

and

contradiction.