Need help in tackling $1^\infty$ form in limits [duplicate]

Question

I was currently studying limits in calculus, and in my book, the following was written to solve the $1^\infty$ indeterminate form:

If $\lim_{x\to0}(f(x))^{g(x)}$, is form of $1^\infty$, then evaluate: $\lim_{x\to o}[f(x)-1]g(x)$. Let the evaluated limit be $\alpha$. Then put $e^\alpha$, which is your answer.

These 'algorithm' steps make no mathematical sense to me and seem some short trick rather than a proper evaluation of limits. The book doesn't care to explain further, than this. So I need help here. Please help me make sense of these steps, if they are correct, or suggest another way to tackle $1^\infty$ forms. Note that I am a high school student, so I won't be able to understand very heavy calculus.

Answer 1

Let me explain the intuition behind the idea. I must say that the formula is quite cool, I haven't seen it before!

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Put $F(x)=f(x)-1$. Recall that $$\ln (1+x)\sim x, \quad x\to 0.$$ Thus also $$\ln(1+F(x))\sim F(x),\quad x\to 0,$$ since $F(x)\to 0$ as $x\to 0$. This can equivalently be expressed as

$$\ln (f(x))\sim f(x)-1,\quad x\to 0.$$

Finally, we obtain $$\lim_{x\to 0}f(x)^{g(x)}=\lim_{x\to 0}\mathrm e^{g(x)\ln(f(x))}=\lim_{x\to 0}\mathrm e^{g(x)(f(x)-1)}=\mathrm e^\alpha,$$ as required.