MATH 1300.03F D **Exam 1**
Friday,
October 8, 1999

NAME:

Student Number:

.**(5 Marks)** Let , .

(*i*) Show that *f* is one to one on .

(*ii*) Find the domain of the inverse function of *f*.

(*iii*) Find a formula for the inverse function.

(*i*) Proof. Assume that *f*(*x*)=*f*(*x*'), .
Then . This implies and *x*=*x*'. Hence, *f* is one to one on .

(*ii*) Let . Then . Hence, the domain of the inverse function is .

(*iii*) , .

. **(4 Marks)** Evaluate the following limits.

(*a*) .

(*b*)

(*c*) .

(*d*) .

. **(6 Marks)** Determine whether each of the following functions
is continuous at *x*=0.

(1)

Soln: We rewrite *f* as follows:

Then, we have

.

This implies . Hence, *f* is continuous at *x*=0.

(2)

Soln: We rewrite *f* as follows:

Hence, we have

This implies does not exist and *f* is not continuous at *x*=0.

Mon Jan 24 11:36:45 EST 2000