Suppose lim xn 7 and lim yn 7 and all yn 0 Determine the f

Suppose lim xn = 7 and lim yn = 7 and all yn=/ 0. Determine the following limits

(i) lim(xn + yn);

(ii) lim(3ynxn)/(yn^2) .

Solution

Given that

lim x_n = 7 and lim y_n = 7

1) lim(x_n + y_n) =   lim x_{n +} lim y_n [ Since , lim_x->a [ f(x) + g(x) ] =  lim_x->a [ f(x) ] + lim_x->a [ g(x) ]

= 7 + 7

= 14

Therefore ,

lim(x_n + y_n) = 14

2 ) lim(3y_nx_n)/((y_n)²) =  lim(3y_nx_n) / lim((y_n)²) [ since , lim_x->a [f(x)/g(x)] = lim_x->a f(x)/lim_x->a g(x) ]

=  [ lim(3y_n) - lim(x_n) ] / lim((y_n)² ) _[lim_x->a [ f(x) - g(x) ] =  lim_x->a  f(x) - lim_x->a g(x) ]  

= [ 3lim(y_n) - lim(x_n) ] / (lim(y_n)²)    [ lim_x->a [cf(x)] = c lim_x->af(x) , c = constant ]

= [ 3lim(y_n) - lim(x_n) ] / (lim(y_n))² [ lim_x->a [ f(x) ]ⁿ = [ lim_x->a f(x)]ⁿ , n = real number ]

= [ 3(7) - 7 ] / (7)²   

= 14/49

= 2/7

Therefore ,

lim(3y_nx_n)/((y_n)²) = 2/7