How do you use the definition of continuity and the properties of limits to show the function is continuous #F(x)= x+sqrt(x-1)# on the interval [1, inf)?

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Answer 1 · 2017-02-09T21:19:00

see below

For #a>1#

We need to show that #F(a)=lim_(x->a) F(x)#. So

#F(a)=a+sqrt(a-1)#

#lim_(x->a) (x+sqrt(x-1) )=lim_(x->a) x + lim_(x->a) sqrt(x-1)#

#=lim_(x->a) x + sqrt(lim_(x->a) (x-1)#

#=lim_(x->a) x + sqrt(lim_(x->a) x-lim_(x->a)1)#

#=a+sqrt(a-1)#

Since #F(a)=lim_(x->a) F(x)=a+sqrt(a-1)# therefore

F is continuous at x = a for every a in #(1,oo)#

We also need to show that #F(1)=lim_(x->1^+) F(x)#

#F(1)=1+sqrt(1-1)=1-0=1#

#lim_(x->1^+) F(x)=lim_(x->1^+) (x+sqrt(x-1))#

#=lim_(x->1^+) x + sqrt(lim_(x->1) (x-1)#

#=lim_(x->1^+) x + sqrt(lim_(x->1^+) x-lim_(x->1^+)1)#

#=1+sqrt(1-1)=1+0=1#

Since #F(1)=lim_(x->1^+) F(x)=1# and F is continuous at x = a for

every a in #(1,oo)# thus F is continuous on #[1,oo)#