Limits and Continuity

There's an infinite discontinuity at $x = c$.

The function has a defined value but may not be continuous at $x = c$.

The limit as $x$ approaches $c$ from both sides is equal to $g(c)$.

The derivative of $g$ at $x = c$ exists and equals zero.

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 $x$.

The integral of $f(x)$ from $a$ to $b$ equals zero.

There are no discontinuities in $f'(x)$ on $(a, b)$.

There are no critical points for $f(x)$ on $(a, b)$.

The function $f''(x)$ exists for all $x$ in $(a, b)$.

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.

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

Use the intermediate value theorem around the value $x=7$ to determine the possibility of discontinuity.

Compare first derivatives on either side of $x=7$ to determine the existence of a common tangent line.

Algebraically solving a system of equations generated from evaluating $f(x)$ in both regimes around the point $x=7$.

Check equality of limits approaching $x=7$ from behind and ahead by plugging in the $x$ value to see if it is continuous.

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.

The function can be traced from the starting point to the ending point without picking up the pencil.

The function is continuous at the starting and ending points of the interval.

The function is continuous at every point in the interval.

The function has no breaks, holes, or discontinuities within the interval.

Confirming that $h'$ has one point where it does not exist within $(2,4)$.

Proving that $h''$ exists everywhere on $[2,4]$.

Ensuring that $h'$ is positive throughout $[2,4]$.

Showing there are no inflection points within $[2,4]$.