To numerically compute the limit $\lim_{x \to 0} \frac{\cos(x) - 1}{x}$, we can use a process that involves calculating the value of the function at successively smaller values of $x$. The idea is to observe how the function behaves as $x$ gets very close to 0.

Numerical Method: Approach

1. Choose a Sequence of $x$ Values: We'll choose $x$ values that get increasingly closer to 0.
2. Compute the Function at These $x$ Values: Calculate $f(x) = \frac{\cos(x) - 1}{x}$ for each of these values.
3. Observe the Trend: As $x$ decreases, observe the trend in the function values to estimate the limit.

I'll perform this computation for smaller values of $x$ to provide a more accurate estimate of the limit.The computed values of $f(x) = \frac{\cos(x) - 1}{x}$ for very small $x$ values are as follows:

$x$	$f(x)$
$10^{-2}$	-0.00500
$10^{-4}$	-0.00005000
$10^{-6}$	-0.000000500
$10^{-8}$	0.000000000
$10^{-10}$	0.000000000
$10^{-12}$	0.000000000
$10^{-14}$	0.000000000

Observation

As $x$ gets smaller and approaches 0, the values of $f(x)$ approach 0. This suggests that:

$\lim_{x \to 0} \frac{\cos(x) - 1}{x} = 0$

This confirms that the limit is indeed 0.

Would you like to explore further details or have any specific questions about the process?

Related Questions:

1. Why does the numerical value of $f(x)$ approach 0 as $x$ gets smaller?
2. How does this numerical approach compare to an analytical approach like L'Hôpital's rule?
3. What are the limitations of using numerical methods for calculating limits?
4. How can numerical methods be used to estimate derivatives at a point?
5. What other types of problems can be solved using a similar numerical approach?

Tip:

When using numerical methods, be mindful of the precision limitations inherent in computing with very small numbers. Sometimes, numerical noise can affect the results, especially when dealing with values close to the machine's floating-point precision limit.