Limits and Continuity

Using L'Hôpital's Rule immediately without any preliminary algebraic manipulation.

Direct substitution of $h=0$ into the expression.

Expanding $(\sqrt{a+h})^n$ using binomial expansion and then taking a limit.

Multiplying by the conjugate of the numerator to rationalize it before taking the limit.

The vertical asymptote becomes horizontal.

The function no longer has a vertical asymptote for any value of $b$.

It does not change; the vertical asymptote remains at $x=a$.

The vertical asymptote moves left if $b<3$ and right if $b>3$.

9/2.

The limit does not exist.

The limit is infinite.

15/4.

Evaluate limits from both sides of $x=5$ to check if they are equal.

Integrate $f(x)$ from $x=4$ to $x=6$ and divide by two.

Multiply $f(x)$ by $-\frac{1}{(5-x)^2}$ and evaluate at $x=5$.

Set $f(5)$ equal to zero since there's a discontinuity at $x=5$.

$\frac{L}{M}$ if M \neq 0

$0$

$LM$

$L+M$

$|v(t)| |_ {t=0} ^ {t=1}$

$|v'(t)| |_ {t=0} ^ {t=1}$

$|\int_0 ^1 v(t) dt|$

$(\max(v(t))-\min(v(t)))|_ {t=0} ^ {t=1}$

9

6

$x^2$

0

The derivative at point $c$, that is, $f'(c)$

Both one-sided limits $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$

The limit as $x$ approaches $b$, i.e., $\lim_{x \to b} f(x)$

The definite integral from $a$ to $b$, $\int_{a}^{b} f(x) , dx$

Undefined

$\infty$

$\frac{5}{4}$

-$\frac{5}{4}$

A line that touches a point on a curve.

A point where a graph intersects the x-axis.

A point where a graph intersects the y-axis.

A line that a graph approaches but doesn’t touch.