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Question
Prove the following properties of limits using the ε-d definition.
Assume that lim as x tends to a of f(x) =k and lim as x tends to a of g(x)= L exist and that c is any constant then prove:
1.lim as x tends to a of [cf(x)]= cK
2.lim as x tends to a of [f(x)+g(x)]=k + L
3.lim as x tends to a of [f(x)g(x)]=KL
4.if f(x) is continuous at x = b and limit as x tends to a of g(x) =b then prove that ;limit as x tends to a of f(g(x))=f(b).
5.assume limit as x tends to a of f(x)=infinity, lim as x tends to a g(x)=L then prove that;lim as x tends to a of [f(x)g(x)]=-infinity if L IS LESS THAN 0
1. It is known that lim(x->a)(cf(x)) = c*lim(x->a)f(x), so that is c*K.
2. The lim(x->a)(f(x)+g(x)) = lim(x->a)f(x) + lim(x->a)g(x) = k + L.
3. The lim(x->a)(f(x)g(x))= [lim(x->a)f(x)][lim(x->a)g(x)] = kL.
4. lim(x->a)f(g(x)) = f(lim(x->a)g(x)) since f and g are continuous, and that is f(b)
since lim(x->a)f(x) = b.
5. This question says lim(x->a)f(x) goes to infinity, and lim(x->a) g(x) = L, so
lim(x->a)[f(x)g(x)] = [lim(x->a)f(x)][lim(x->a)g(x)] = (infinity)L = infinity.
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