Why were limits introduced in calculus?

Question

I have got the intuitive of the limits but why were they introduced in calculus, are they really helpful to us?

Answer 1

Of course they are helpful in calculus. Calculus cannot be formulated without limits, as essentially, calculus is the theory of infinitesimals as $\Delta x\to 0$.

You define the differential as a limit: $$f'(x)=\lim_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$$ You define the integral as a limit: $$\int^b_af(x)\operatorname{d}x=\lim_{\Delta x\to 0}L_n=\lim_{\Delta x\to 0}R_n$$ Where $L_n,R_n$ are the left and right Riemann sums respectively. You will also encounter limits in topology.