Let f (x) be continuous on [0, ∞), and if x > 0, 0

Let f (x) be continuous on [0, ∞), and if x > 0, 0

We only need to prove that: F has an increasing bound
in fact,
1) If f (x) is increasing, the derivative is greater than 0;
2) F (x) has an upper bound
Use f (x) = f '(s) to integrate from 1 to x, plus f (1)
Because f '(x)

When x tends to zero, x times cos1 / X limit?
It's infinitesimal. Why

X tends to 0, 1 / X tends to infinity, cos1 / X is a bounded quantity, - 1 ≤ cos1 / X ≤ 1
So according to the nature of infinitesimal four operations
Infinitesimal multiplied by bounded variable = infinitesimal
Namely
When x tends to zero, Lim xcos1 / x = 0
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The limit limx (sqrt (x ^ 2 + 100) + x) tends to be negative infinity

limx(sqrt(x^2+100)+x)
=lim100x/(sqrt(x^2+100)-x)
=lim100/(-sqrt(1+100/x^2)-1)
=-50