Limits and Continuity

$\frac{1}{4}$

0

-4

Undefined

Call the function $f'(a)$.

Apply L'Hopital's Rule.

Expand $e^{xt}$ using McLaurin series then find the limit as $x$ approaches $a$.

Use polynomial long division.

Multiplying both numerator and denominator by its conjugate pair.

Differentiating both top and bottom as a preparation for applying L'Hôpital's Rule if needed.

Expanding powers of binomials in both numerator and denominator before simplifying terms.

Factoring $x^3 +27$ as $(x+3)(x^2 -3x +9)$ and canceling one $(x+3)$ term.

$\sqrt{(X)}$

$X$

Zero

$|X|$

$\frac{L}{M}$

$L + M$

$L \cdot M$

The limit does not exist.

0

1

Undefined

-1

Factor out a common term from numerator and denominator before simplifying.

Direct substitution which results in an indeterminate form (division by zero).

Rationalize the numerator by multiplying by its conjugate.

Apply L'Hopital's Rule directly without simplifying first.

.3 +1

-1

Does not exist

Undefined because division by zero is involved

$k = 2$

$k$ can be any real number

$k = -2$

$k = 0$

$\frac{3}{5}$

$\infty$

$-\frac{3}{5}$

Undefined