[tex]\it ln\dfrac{2x}{x^2+x} = ln\left(2\cdot\dfrac{x}{x^2+x}\right) = ln2 +ln\dfrac{x}{x(x+1)} = ln2 +ln\dfrac{1}{x+1}
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\lim \limits_{x \to 0} ln \dfrac{1}{x+1} = ln \lim \limits_{x \to 0} \dfrac{1}{x+1} = ln1 = 0
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\lim \limits_{x \to 0} ln\dfrac{2x}{x^2+x} = \lim \limits_{x \to 0} \left(ln2 +ln\dfrac{1}{x+1}\right) = ln2 + \lim \limits_{x \to 0} \left(ln\dfrac{1}{x+1}\right) = [/tex]
[tex]\it = ln2 +0 = ln2.[/tex]
