The value of $\lim_{t \to 0^+}\frac{(p-2)t^{p-2} + (q-2)t^{q-2}}{t^{p-2} + t^{q-2}}$?

Question

Let $ f \colon \mathbb (0,\infty) \rightarrow \mathbb R $ defined by

$$f(t) = \frac{(p-2)t^{p-2} + (q-2)t^{q-2}}{t^{p-2} + t^{q-2}}$$

for all $t >0 $ with $ 1 <q<p< \infty $.

I need to calculate this limit:

$$\lim_{t \to 0^+}\frac{(p-2)t^{p-2} + (q-2)t^{q-2}}{t^{p-2} + t^{q-2}}.$$

I would really appreciate your help with this problem thank you.

Answer 1

Hint. Note that $p-q>0$ and for $t>0$, $$f(t)=\frac{(p-2)t^{p-q} + (q-2)}{t^{p-q} + 1}$$