Limit of a definite integral from x0 to x, as x approaches x0

The integral from x₀ to x of f(t) dt, assuming it exists, is an antiderivative of f(x) by the FTC, in particular it is a continuous function of x (since it is differentiable). So its limit as x goes to x₀ is the same thing as the integral from x₀ to x₀, which is zero.
Assuming your function is integratable/ differentiable, then yes. You would get something like like *lim [F(x) - F(a), a->x] = F(x) - F(x) = 0*.

Edit: Reasoning: the limit of a difference is equal to the difference of the limits. So as *lim "def integral"* yields you something like *lim [F(b) - F(a)]* you can rewrite it as *lim [F(b)] - lim [F(a)]*.
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