Limits and Continuity

6 units/time

-9 units/time

2 units/time

12 units/time

$\lim_{c \to x^-} g(c)$

$\lim_{c \to x^+} g(c)$

$\lim_{x \to c^-} g(x)$

$\lim_{h \to c} g(h)$

The function b(h) must not be differentiable at h=0 because if it were, its derivative would cancel out with that of a(h).

Continuous extension guarantees that taking the limit is sufficient without additional conditions on derivatives being met at h=0.

The functions $a(h)$ and $b(h)$ must intersect at h=0 for their ratio to have a meaningful limit as h approaches zero.

Derivatives of $a(h)$ and $b(h)$ must exist around h=0 and their limiting ratio must also equal L after application.

Differentiability depends on whether a graph has an endpoint which continuous functions do not have.

Continuous functions are always differentiable since they have no breaks or holes.

A continuous function may have sharp corners or cusps where there are no defined tangents.

Continuity means that all vertical tangents are possible making differentiability certain.

Factor $g(t)$ to identify possible maxima within the interval.

Apply L'Hôpital's Rule to solve for limits approaching maxima.

Use integration from $t=2$ to $t=5$ to find areas under curves.

Evaluate $g(t)$ at critical points and endpoints then compare values.

0 ft/s

16 ft/s

64 ft/s

32 ft/s

As $n$ increases without bound, $p(n)$ gets arbitrarily close to zero.

There must be some point beyond which $p(n) > 0$ no matter how large $n$ gets.

$p(n) = 0$ for all sufficiently large $n$.

The function $p(n)$ decreases monotonically for all $n$.

$27$

$-6$

$9$

$18$

-16 ft/s^2

32 ft/s^2

-32 ft/s^2

16 ft/s^2

The total amount of interest gained by the bank account after 20 days.

The total amount of money in the bank account after 20 years.

The total amount of money in the bank account after 20 months.

The total amount of interest gained by the bank after 20 years.