calculus - Can we still use L'Hospital's rule on limit that does not exists?

I have the following limit and I must find the value of it:

$\lim\limits_{x\to0}{\frac{\ln(\tan 2x)}{\ln(\tan 3x)}}$

The corrected version of this exercise provided by my teacher says it equals 1 and that's the value I got using L'Hospital's rule.

However, from what I know of limits, the limit does not seem to exists since $\lim\limits_{x\to{0^+}{}}{{\ln(x)}}$ exists but not $\lim\limits_{x\to{0^-}{}}{{\ln(x)}}$. Thus $\lim\limits_{x\to{0}{}}{{\ln(x)}}$ does not exists either.

Am I wrong or is the corrected version wrong? Also, I tried to use online websites to find the limit's value and they also find 1 as well for the first limit as for the one I used as an example.

Answer

If a function $f$ is a real-valued function whose domain is some subset $A$ of the real line for which $x_0$ is a limit point of both $(-\infty,x_0)\cap A$ and $(x_0,\infty)\cap A,$ then we can conclude that $\lim_{x\to x_0} f(x)$ exists if and only if $\lim_{x\to x_0^-}f(x)$ and $\lim_{x\to x_0^+}f(x)$ exist and are equal to each other.

In general, if we define limits as follows:

Suppose $f$ is a real-valued function whose domain is some subset $A$ of the real line, and suppose $x_0\in\Bbb R.$ We say the limit of $f(x)$ as $x$ approaches $x_0$ exists if (and only if) the following hold:

• $x_0$ is a limit point of $A,$ and




• there is some $L\in\Bbb R$ such that, for any $\epsilon>0$ (no matter how small), we can find some $\delta>0$ such that whenever $x\neq x_0$ is within $\delta$ of $x_0,$ we have $f(x)$ within $\epsilon$ of $L.$

We denote this $L$ (if it exists) by $$\lim_{x\to x_0}f(x).$$

As for one-sided limits, we define them as follows:

Suppose $f$ is a real-valued function whose domain is some subset $A$ of the real line, and suppose $x_0\in\Bbb R.$ We say the limit of $f(x)$ as $x$ approaches $x_0$ from the left exists if (and only if) the following hold:

• $x_0$ is a limit point of $A\cap(-\infty,x_0),$ and




• there is some $L\in\Bbb R$ such that, for any $\epsilon>0$ (no matter how small), we can find some $\delta>0$ such that whenever $x

We denote this $L$ (if it exists) by $$\lim_{x\to x_0^-}f(x).$$ We similarly define the existence of the limit of $f(x)$ as $x$ approaches $x_0$ from the right, and denote it (if it exists) by $$\lim_{x\to x_0^+}f(x).$$

We can abuse the notations above to indicate increase/decrease without bound as $x$ approaches $x_0$ from two sides or just one. If $x_0$ is a limit point of both $A\cap(-\infty,x_0)$ and $A\cap(x_0,\infty),$ then $f(x)$ increases (decreases) without bound as $x$ approaches $x_0$ if and only if it does so as $x$ approaches $x_0$ from both the left and the right. However, if $x$ is only a limit point of either $A\cap (-\infty,x_0)$ or $A\cap(x_0,\infty),$ we can still talk about $\lim_{x\to x_0}f(x),$ as long as we can talk about the one-sided limit that actually makes sense.

In your particular cases, $\lim_{x\to x_0^+}f(x)=\lim_{x\to x_0}f(x),$ since neither numerator nor denominator is defined for $x\le0.$ So, you can still apply the rule.