Limits and Continuity

p(x) and q(x) are both continuous functions across their domains independent of each other except where $q(x)=0$.

$\frac{p'(x)}{q'(x)}$ should exist and be finite across their shared domain exclusive of points where $q'(x)=0$.

p(x) must intersect q(x) at least once in their domains excluding where $q(x)=0$.

p(x) must have a higher degree polynomial than q(x).

A jump discontinuity where there is a sudden change in function values within the interval.

An infinite discontinuity where the function approaches infinity within that interval.

An indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ at a point in the interval.

A removable discontinuity where there is a hole in the graph of the function on that interval.

No, it is not continuous over the interval $(-1, 3)$.

Yes, it is continuous over the interval $(-1, 3)$.

The continuity of $m(x)$ cannot be determined based on the given information.

It is continuous for all values of $x$.

Neither continuous nor differentiable

Both continuous and differentiable

Differentiable but not continuous

Continuous but not differentiable

The limit as $x$ approaches $a$ of $f(x)$ exists, but is not equal to $f(a)$.

The limit as $x$ approaches $a$ of $f(x)$ equals $f(a)$.

The function has a vertical asymptote at $x = a$.

The limit as $x$ approaches $a^+$ of $f(x)$ equals the limit as $x$ approaches $a^-$ of $f(x)$, but both do not equal to $f(a)$.

$\lim_{x \to b^-} g(x) = \lim_{x \to b^+} h(x) = g(b) = h(b)$

$\lim_{x \to b} g'(x) = \lim_{x \to b} h'(x)$

Both pieces have vertical asymptotes at x=b.

$\lim_{x \to b^-} g'(x) = \lim_{x \to b^+} h'(x)$

The derivatives $\sin z$ & $\tan z$ never both reach zero simultaneously within the given range.

When $\tan z$ takes value one.

When $\sin z$ reaches maximum value.

When derivative $\cos z$ doesn't exist anywhere within the interval.

k=-2

k=1

k=0

k=3

There's a vertical asymptote at $t=c$

$\lim_{{t \to c^-}} g(t) \neq \lim_{{t \to c^+}} g(t)$

$\frac{dg}{dt}$ exists for all points except at $t = c$

$\lim_{{t \to c}} g(t) = g(c)$

Attempting polynomial long division on $(\sin(x))^x$ prior to evaluating limits.

Using trig identities alone to transform $(\sin(x))^x$ before taking limits.

Applying logarithmic differentiation followed by l’Hôpital’s rule.

Conducting synthetic division involving terms of $(\sin(x))^x$, then assessing limits.