2. (3 pts total) We know that tangent lines approximations typically get worse the further you go from the point $(a, f(a))$. However, this is not always the case. In fact, sometimes, the tangent line can even touch the function at more than one point.
Recall that the error of a tangent line estimate is
$E(x) = f(x) - f'(a)(x - a) - f(a)$
(a) (2 pts) Find a function $f$ and values $a < b < c$ such that $|E(b)| > |E(c)|$. In other words, the error at $b$ is greater than the error at $c$, even though $c$ is farther away from $a$. Show why your example works.
(b) (1 pt) Which function property do examples from part (a) share? Hint: what property should $E(x)$ satisfy?
(c) (bonus 1 pt) As a challenge, find a nonlinear function $f$ and point $a$ such that for the tangent line at $a$, $\lim_{x\to\infty} E(x) = 0$. In other words, the tangent line approximation gets even better the further you are from $a$.