By using the definition of limit of functions, show that

[Guest]

By using the definition of limit of functions, show that
[tex]lim_{x->0} x.\frac{1}{x} = 1[/tex]

[Guest]

The lim is quite simple because
[tex]1*\frac{1}{x} = 1[/tex]
so
lim 1 = 1
no matter of x

[AliRaza08]

The lim can be quite effortless because
1∗1x=1
so
lim 1 = 1
no matter connected with x




Kashmiri Butt

[Guest]

By using the definition of limit of functions, show that
[tex]lim_{x->0} x.\frac{1}{x} = 1[/tex]

Well, since you say "by using the definition of limit of functions", I would start stating the "definition of limit of functions"! It is "[tex]lim_{x\to a} f(x)= L[/tex] if and only if, for every [tex]\epsilon> 0[/tex] there exist [tex]\delta> 0[/tex] such that if [tex]|x- a|< \delta[/tex] then [tex]|f(x)- L|< \epsilon[/tex]." Here, a= 0, f(x)= x(1/x)= 1, and L= 1 so that you want to prove "given [tex]\epsilon> 0[/tex] there exist [tex]\delta> 0[/tex] such that if [tex]|x|< \delta[/tex] then [tex]|1- 1|= 0< \epsilon[/tex]. But 0 is less than any positive number so it doesn't matter what x is and we can take [tex]\delta[/tex] to be anything.

Did you not realize that x(1/x)= 1 for all (non-zero) x or are we misunderstanding what you mean by [tex]x.\frac{1}{x}[/tex]?