Example question using Taylor series with Limits
How do you go about solving limits like the one shown above? Why is it not correct to express the taylor series in terms of y instead of x, and then multiply the taylor series by $\sqrt{y}$, followed by subtracting y? Like so:
$\sqrt{y^2+y} = (1+\frac{1}{2}y-\frac{1}{8}y^2)\sqrt{y} - y$
When $y$ tends to $+\infty$, $$\sqrt{y^2+y} - y = y \left( \sqrt{1+\frac{1}{y}}-1\right) = y \left( \frac{1}{2y} + o \left( \frac{1}{y}\right)\right) = \frac{1}{2} + o(1)$$
So the limit is $\frac{1}{2}$.