Cannot find limit

Question

I was solving an improper integral, but I am stuck in here $$\lim_{b\rightarrow +\infty} b^{1-\theta} $$ could you write, please, how to solve this limit?

Thank you

Answer 1

Given

$$\int \limits_{k}^{\infty}\frac{1}{y}^{\theta+1}y{\theta}k^{\theta}dy$$

note that

$$\frac{1}{y}^{\theta+1}y{\theta}k^{\theta}\sim \frac{{\theta}k^{\theta}}{y^{\theta}}$$

thus by limit comparison test with $\int \frac{1}{y^{\theta}}$ the integral converges for $\theta >1$

Answer 2

We can rewrite it as $$b^{1-\theta}=e^{(1-\theta)\log(b)}$$ And as $b\to \infty$, $\log(b) \to \infty$.
If $1-\theta<0$, then it will go to $0$.
If $1-\theta =0$, then it will go to $1$.
If $1-\theta >0$, then it will go to $\infty$.