I have no idea how to start. Tried L`Hospital and Newton Leibnitz rule but was stuck. Plz clarify

It sounds like you're grappling with a challenging problem involving limits, possibly one that requires the application of L'Hôpital's Rule or the Fundamental Theorem of Calculus (Newton-Leibniz). Let's break this down step by step to clarify how to approach such problems effectively.

Understanding L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms, specifically 0/0 or ∞/∞. The rule states that if you have a limit of the form:

lim (x → c) f(x)/g(x) = 0/0 or ∞/∞

then you can differentiate the numerator and the denominator separately:

lim (x → c) f'(x)/g'(x)

provided that this new limit exists. This is often a game-changer when you're stuck.

Applying L'Hôpital's Rule

Let’s say you have a limit problem like:

lim (x → 0) sin(x)/x

This limit is of the form 0/0. To apply L'Hôpital's Rule, differentiate the numerator and denominator:

• f(x) = sin(x) → f'(x) = cos(x)
• g(x) = x → g'(x) = 1

Now, the limit becomes:

lim (x → 0) cos(x)/1 = cos(0) = 1

This shows how L'Hôpital's Rule can simplify the problem significantly.

Newton-Leibniz Theorem Overview

The Fundamental Theorem of Calculus connects differentiation and integration, providing a way to evaluate definite integrals. If you have a function F that is the antiderivative of f, then:

∫ from a to b f(x) dx = F(b) - F(a)

This theorem is particularly useful when you need to find the area under a curve or evaluate limits involving integrals.

Example of the Fundamental Theorem

Consider you need to evaluate:

lim (x → 0) (∫ from 0 to x t^2 dt)/x

First, find the integral:

∫ t^2 dt = (1/3)t^3

So, the definite integral from 0 to x is:

(1/3)x^3

Now, substitute this back into the limit:

lim (x → 0) (1/3)x^3/x = lim (x → 0) (1/3)x^2 = 0

This shows how the Fundamental Theorem can be applied to evaluate limits involving integrals.

Final Thoughts

When you're stuck on a limit problem, always check if it fits the criteria for L'Hôpital's Rule or if you can apply the Fundamental Theorem of Calculus. Both methods can simplify complex problems significantly. If you have a specific example you're working on, feel free to share it, and we can tackle it together!