Limits and Continuity

$\infty$

Undefined

$0$

Does not exist

The limit equals one as x approaches the vertical asymptote value.

The derivative of g(x) goes to zero as x approaches the vertical asymptote value.

The limit does not exist or is infinite as $x$ approaches the vertical asymptote value.

The limit equals zero as x approaches the vertical asymptote value.

The slope of f at x=a is undefined.

There may be a vertical asymptote at x=a.

The graph crosses through x=a.

There’s a horizontal asymptote at y=-∞.

Negative infinity ($-\infty$)

Infinity ($\infty$)

Does not exist (DNE)

Zero (0)

The function is undefined.

The function approaches negative infinity.

The function approaches 0

The function approaches positive infinity.

Calculating the limit as $t$ approaches $s$ from both sides to assess whether it is infinite or does not exist hints towards vertical asymptote presence.

Adopting a numerical sequential approach, substituting successively close $t$ values around $s$-predicted zone to catch incremental $y$-value escalation suggesting vertical asymptote occurrence.

Algebraically determining $s$ values causing the denominator zero allows identification of vertical asymptotes precisely without graphical tools requirement.

Implementing derivative techniques to evaluate behavior near $s$-determined points, understanding potential inflection influence over asymptotic tendency.

$-\infty$

$\frac{1}{2}$

$+\infty$

$0$

The function approaches positive infinity.

The function approaches negative infinity.

The function oscillates without bound.

The function remains constant.

9

Undefined

3

0

The function approaches positive infinity.

The function approaches 0.

The function approaches negative infinity.

The function approaches 1.