I am being asked to work out:
$$\lim_{x \rightarrow \infty} \Big( \sqrt{x^{2}+5x} – x \Big)$$
My thinking is — OK, so $x$ is some very big positive No, so I can rewrite as
$$\lim_{x \rightarrow \infty} \Big( \sqrt{x^{2}+5x} – \sqrt{x^{2}} \Big)$$
I can see that $5x$ will ensure that I approach infinity.
However the actual answer is to use Taylor series approximation around zero (which in itself seems questionable, considering $x$ is approaching infinity) to arrive at an answer of $2.5$.
Where am I going wrong in my thinking. Am I?
Best Answer
Hint: A different way is to rationalise the numerator: $$\sqrt{x^2+5x}-x=\dfrac{5x}{\sqrt{x^2+5x}+x}=\dfrac{5}{\sqrt{1+\frac{5}{x}}+1}$$