Limits and Continuity

$\frac{1}{4}$

0

-4

Undefined

11

9

Undefined

10

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.

Absolute value of x, $|x|$.

Unit vector $\frac{x}{|x|}$.

Converges to a constant because of dominance of x in terms.

Infinity, due to larger rate of growth in the numerator.

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.

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.

First step should include checking if the function is continuous at the point before proceeding with algebra.

Initiating process with application of L'Hôpital's Rule can provide a direct pathway to a solution.

Adopt the strategy of multiplying and dividing by conjugate in order to resolve sin, since this action will lead to the removal of the sine expression.

Approach should begin transformation into polar coordinates for simpler computation.

$\sqrt{(X)}$

$X$

Zero

$|X|$

0

None of the above

Infinity

Undefined

1

Undefined

1/2

2