calculus

Question

what is difference between this and in questions if asked to find derivative or check differentiable or not, which formula should i use?

1) $lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$

 

2) $lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a}$

 

Comments

both of them is same because if you replace x by x+a in second eq we will automatically get eq 1.you can use any one of them but for  more simplicity you can use eqn (1)

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how can we replace x by x+a ?

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let f(x) be x-5 and consider the limit x->1 here a is 1  by replacing x to x+1 then lim become x->0 we must replace x+1 in f(x) now put the value we will get the answer 1 from both method.
if lim x->h then we can replace lim x+h->h where h is small quantity after that it become lim x->0.you may refer class 12 ncert book of mathematics.

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Both are actually same.

lets take the second one,

$\lim_{x->a} \frac{f(x)-f(a)}{x-a}$

now lets assume ,

$h=x-a$ so as $x->a$ the $h->0$

now , x=h+a , substituting this in the formula,

$\lim_{h->0}\frac{f(a+h)-f(a)}{h}$  so, it is the first expression