Let f: SR+ R

be given by f(x)=(x^1/2 -1)/(x1)

(a) derive Lim _x->1 f(x)

(b) Is f continuous at the point x=1? Briefly explain.

(c) Definition 2.1 : Let S and T R and f: S T

Let x0 be an adherent point of S. The function f(x) converges to the limit L as x tends to x0 if, for every >0, there exists a ()>0 such that |x -x0| () |f(x)-L| .

According to Definition 2.1 what do you need to show to prove that f is continuous at x0 S.

(d) Show that |[1/(x^1/2+1)]-[1/(x0^1/2+1)]|1/2-x^1/2|

(e) Use the result in (d) to prove that f is continuous at x0 S. Hint: Since x0^1/2 0,

it follows that x^1/2 x0^1/2 x^1/2+ x0^1/2.

Question 1 (17 marks) Let f : S C R+ > R be given by _ J? 1 X) _ X _ 1 (a) Use Theorem 2.2 to derive Lim f(x). x41 (2 marks) (b) Is f continuous at the point x = 1? Briey explain. (2 marks) (c) According to Denition 2.1 what do you need to show to prove that f is continuous at JC() 6 S. ( 2 marks) (d) Show that l 1