Any tips to solve these limits ?

Question

Concidering that f  is continious over the interval [0;1] 

lim n→∞ ∫₀¹ ( f (x) /x+n ) dx

lim n→∞ ∫₀¹ (xⁿ f (x) ) dx

lim n→∞ ∫₀¹ (f (x)/1+nx) dx

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lim n→∞ ∫₀ⁿ (xⁿ e^-nx ) dx

Answer 1

I am not sure I can solve all of these for you, but I can give you one big hint:

since f is continuous on the interval, it is bounded. Call the bound B.

In the first one the integral is less than B*∫dx/(x+n) = B*ln[(n+1)/n] = 0.in the limit

In the 2nd one the integral is less than B∫xⁿ dx = B*[1/n+1] = 0 in the limit

In the 3rd one the integral is less tan (B/n)*∫dx/{x+(1/n)] = (B/n)ln](n+1)/n] = 0 in the limit

I will have to think about the 4th one & if I can give you further help, I will get back to you.