find $\lim _ {y\rightarrow + \infty } \int _ { 1 } ^ { 2 } \frac { \ln ( x + y ) } { \ln \left(x^{2}+y^{2} \right) } d x$

Question

I need to find $$\lim _ {y\rightarrow + \infty } \int _ { 1 } ^ { 2 } \frac { \ln ( x + y ) } { \ln \left(x^{2}+y^{2} \right) } d x$$ Any hints? PS. i tried to convert problem to solving improper integral. But nothing good happened

Answer 1

As $y\to\infty$ the addition of $x/x^2$ does not greatly affect the values of the arguments of the functions, so the integral tends to $$\int_1^2\frac{\ln{(y)}}{\ln{(y^2)}}\mathrm{d}x=\int_1^2\frac{\ln{(y)}}{2\ln{(y)}}\mathrm{d}x=\int_1^2\frac12\mathrm{d}x=\frac12(2-1)=\frac12$$

Answer 2

By dominated convergence, you may move the limit operand inside, giving $\int_1^2\frac{dx}{2}=\frac12$.