limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Question

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers?

I already know how to use the L' Hopital's rule and the strategy on using the limit $L=\lim_{x\rightarrow\infty}\ln(f(x))$ then substituting it back to find $\lim_{x\rightarrow\infty}f(x)= e^L$.

But how about this one? I tried to use the techniques but I can't find a mathematically "legal" way to do it, (abiding the rules of mathematics with a legitimate formula based on the theorems).

I've seen a trend on it using these graphs, but can you kindly show me how to solve it using the theorems of mathematics?

$a=b, c>d$

$a>b, c>d$

$b>a, c>d$

$a=b, c>>d$

$a=b, d>c$

Answer 1

Let $a,b,c,d$ be positive; let also $L=\max (a,b,c,d)$.

Now consider $\lim_{x\to+\infty}\frac{a^x+b^x}{c^x+d^x}=\lim_{x\to+\infty}\frac{(a/L)^x+(b/L)^x}{(c/L)^x+(d/L)^x}$ and conclude.

The limit $x\to-\infty$ is done similarly.