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If we are to prove a limit with a differentiable function $f(x)$,we know that:$$\lim_{x \to a} f(x) =\lim_{x \to a} f(a)+f'(a)(x-a)$$

So can we rigorously use this linear approximation in our proof instead of the original function??

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If $f$ is differentiable at $x=a$, it is continuous at $x=a$, so you are assuming what you want to prove. That being said, if you do use this, as long as you are careful. Specifically, you can not just use the approximation $$f(x) \approx f(a)+f'(a)(x-a)$$ but instead include the error terms as such: $$f(x) = f(a)+f'(a)(x-a)+O((x-a)^2)$$ When you use this, you will show that this final term goes to zero in the limit, so it works out in the end. The only potential issue is that the error term depends on the second derivative,

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    I know differentiability implies continuity ,but I want to provide a direct proof instead of using intermediary theorems... – 2017-01-06
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    Right, that's what I suspected,so I included the steps after the initial statement. – 2017-01-06