Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^\prime(x)$ exists, show that $f$ is differentiable at $a$

Question

I have some troubles in trying to prove the following result and possibly need some guidance:

What I tried. I started the proof as follows
If $x < b$, then $f$ is differentiable on $(a, x)$. So by the Mean Value theorem, $$ f^\prime(c) = \frac{f(x) - f(a)}{x - a}. $$ Now we want to show that $\lim\limits_{{x \to a}} \frac{{f(x) - f(a)}}{{x - a}}$ exists (and is equal to $L$), so we should estimate the difference $$ \left| \frac{f(x) - f(a)}{x - a} - L \right|=|f^\prime(c) - L| $$ From now on I am not sure on what to do: any help would be very much appreciated.
A note. My professor specifically asked to prove the statement by using the Mean Value Theorem, so I can't use any solution revolving around L'Hopitals Rule.

We haven't learnt lhopitals rule in my real analysis class yet, my professor specifically asked us to prove it using the mean value theorem, sorry for the confusion ill edit my post — Muhammad Abdurrahman Ullah, Dec 01 '23 at 01:55
In that case, what you ve written is fine. Just send $x\searrow a$ and you are done — Andrew, Dec 01 '23 at 02:08
Assuming you are working with the formal $\varepsilon$-$\delta$ definition of the limit, you just need to let $\varepsilon > 0$, get some $\delta > 0$ such that $a < c < a+\delta$ implies $|f'(c) - L|<\varepsilon$, and then let $x$ be such that $a <x < a+\delta$. — Brian Moehring, Dec 01 '23 at 02:12
Yes this was my approach to the question, but im having difficulty finding the proper delta such that $|f'(c) - L| < \epsilon $ — Muhammad Abdurrahman Ullah, Dec 01 '23 at 02:22
I was wondering if you could elaborate how that shows the function is differentiable at a, I'm not quite understanding. — Muhammad Abdurrahman Ullah, Dec 01 '23 at 02:31
See the question Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. for various answers. — Sarvesh Ravichandran Iyer, Dec 01 '23 at 04:15
As for "how do we find $\delta$?" - Just use the assumption that the limit of $f'(x)$ exists as $x\to a$. — Brian Moehring, Dec 01 '23 at 09:02

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^\prime(x)$ exists, show that $f$ is differentiable at $a$

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