Numerical Approximation | Differential Calculus

1. Look at the table below representing $f(x)$. What is the best estimate for $\lim_{x \to 2} f(x)$?

x	1.9	1.99	1.999	$\to 2 \leftarrow$	2.001	2.01	2.1
f(x)	4.7	4.97	4.997	Undef.	5.003	5.03	5.3

A) Undefined B) 2 C) 5 D) 0

2. You construct a table for $g(x)$ as $x \to 4$. From the left, the values of $g(x)$ are $7.9, 7.99, 7.999$. From the right, the values of $g(x)$ are $2.1, 2.01, 2.001$. What can you conclude about $\lim_{x \to 4} g(x)$?

A) The limit is 8. B) The limit is 2. C) The limit is 5 (the average). D) The limit does not exist.

3. Why do we evaluate values very close to $a$ (like $a \pm 0.001$) instead of just evaluating $f(a)$ directly when finding a limit numerically?

A) Because evaluating $f(a)$ directly is often undefined (e.g., division by zero), and a limit describes the behavior near the point, not exactly at it. B) Because decimals are more accurate than integers. C) Because $f(a)$ is always guaranteed to be 0. D) Because evaluating $f(a)$ is cheating.