4.3 THE LIMIT CONCEPT (OPTIONAL) 145depends on E; specifically, the smaller E is chosen, the larger will be corresponding
values of N that can be chosen.
(a) In the case where the graph off is a continuous curve, what is the relation-
ship between the line y = L and the graph off, when lirn,,, f(x) = L?
(b) Evaluate these limits (Hint: Let x = l/t, simplify, and let t -, 0):
(i) lim,,, (x - 3)/(x + 4)
(ii) limX,,(x+6)/(x2- 10)
(iii) lirn,,, (xZ - 2x + 3)/(2x2 + 5x - 3)
(iv) lirn,,, (xZ - 6x - 7)/(x + 2)
(c) Write a formal definition of lirn,, -, f(x) = L.
(d) *(i) Write a formal definition of lirn,,, f(x) = m.
(ii) In a case where f is continuous in an open interval containing a, but
not at a itself, and lim,,, f(x) = m, what is the relationship between the
graph off and the line x = a?