Evaluate the limit:$$\lim_{x\to \infty}\frac{(x+1)^1+(x+2)^2+(x+3)^3+........(x+100)^{100}}{x^{10}+10^{10}}$$
In my book they taken the higher power out i mean $x^{100}$ out and then got the answer but my doubt is how can he take individual limits without knowing the continuity of the function. I got bit confused by this About the computation of the limit $ \lim_{x \to \infty} \frac{1^{99} + 2^{99} + \cdots + x^{99}}{x^{100}} $
So please explain.
Divide the numerator and the denominator by $x^{10}$, which is the highest power there.
Then look at the limit of the denominator as $x \to \infty$. That should be a number.
Next look at the limits of the first, second, ..., tenth term in the numerator. These are all numbers. Work out each term until you see the pattern. Pay special attention to the tenth term.
Then work out the eleventh term and a few of the following terms.
At that point you should see the answer.
Continuity is not an issue here. Since $x \to \infty$, you can always assume that $x > 1$ and everything is defined.