Post by ææ¯å¡å°¼è·é½ä¸f(x) is differentiable,
If lim f'(x) = oo
x ->oo
prove or disprove that lim f(x) = oo.
x->oo
We can find M such that f'(x) > 1 for x ≧ M > 0.
Let H(x)= (f(x)-f(M))-(x-M). Then H(M) = 0 and H'=f'-1 > 0 for x ≧ M.
Hence, H is increasing, that implies H > 0 for x≧M.
Given any N > 0, let S = M + |f(M)| + N.
If x > S, H(S) = f(x)-f(M) - x + M < f(x)-f(M) - M - |f(M)| - N + M
= f(x) - (f(M) + |f(M)|) - N.
Since H(S) > 0, f(x) - (f(M)+|f(M)|) -N > 0; hence,
f(x) > N + (f(M)+|f(M)|) ≧ N.
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