Friday, 26 August 2016

limits - Evaluate limxtoinftyln(x3+x)




Evaluate lim




I was wondering if I can solve this limit in this way:
\lim_{x \to -\infty} \ln(-x^3+x)=\lim_{x \to -\infty} \ln\left[x^3\left(1+\frac{1}{x^2}\right)\right].




At this point, I just considered \ln(x^3) because 1 doesn't make any difference and 1/x^2 tends towards 0.
So, the result will be 0. And I found it because I know the graph of the logarithm of x to the power of an odd number. So, my second question is, is it possible to understand what the result of \lim_{x\to - \infty} \ln(x^3) algebraically without thinking of the graph?



Any suggestion and help will be appreciated.
Thank you in advance and have a good day :)


Answer



The correct way is



\lim_{x \to -\infty} \ln(-x^3+x)=\lim_{x \to -\infty} \ln\left[-x^3\left(1-\frac{1}{x^2}\right)\right]=\lim_{x \to -\infty} \left[\ln -x^3+\ln\left(1-\frac{1}{x^2}\right)\right]=\\=\lim_{x \to -\infty} \ln -x^3+\lim_{x \to -\infty} \ln\left(1-\frac{1}{x^2}\right)=+\infty+0=+\infty




indeed



\lim_{x \to -\infty} \ln -x^3=\lim_{x \to -\infty} 3\ln -x=\lim_{y \to +\infty} 3\ln y=+\infty


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