Contextual Applications of Differentiation

When the limit is already determinate

When the limit is approaching zero

When the limit is a finite value

When the limit involves trigonometric functions

Verifying that h does not equal zero within any trigonometric identities present in either sine or tangent functions.

Proving that sine grows faster than tangent as h approaches zero through their Maclaurin series expansions.

Establishing continuity on both numerator and denominator separately around h equals zero.

Confirming that both sine and tangent approach zero as h approaches zero creating an indeterminate form ($\frac{0}{0}$).

Expanding n! into its components and then directly applying L’Hôpital’s Rule repeatedly.

Applying direct substitution first, since both numerator and denominator approach infinity.

Rewriting $\sqrt[n]{n!}$ as $(n!)^{1/n}$ and recognizing a growth rate comparison between polynomial and exponential functions.

Factoring out terms from the factorial in the numerator to simplify before applying L’Hôpital’s Rule.

Only when $k = \frac{1}{2}$

No such value exists; it’s an indeterminate form.

Any real number except zero.

Only when $k = \pm\sqrt{2}$

The equation rearranged to a rational expression

The derivative of the function

The limits of $f(x)$ and $g(x)$ separately equal to the required parameters

The original equation with all terms cancelled out

The limit does not exist because it approaches different values from either side.

The limit equals $\frac{-1}{2\sqrt{x}}$

The limit equals $\frac{1}{2\sqrt{x}}$

The limit equals $\sqrt{x}$

Zero (0)

Infinity ($\infty$)

One (1)

-Infinity ($-\infty$)

Multiply top and bottom by conjugates to simplify the expression.

Integrate both top and bottom functions before evaluating.

Take derivatives of top and bottom functions and re-evaluate the limit.

Add constants to both numerator and denominator until it resolves.

The limit is $\frac{1}{2}$.

The limit does not exist.

The limit is $0$.

The limit is $1$.

The derivatives of both $e^x$ and $-e^{-x}$ with respect to $x$.

The integral of $\frac{e^x - e^{-x}}{x}$ with respect to $x$.

The second derivative of $e^x - e^{-x}$ with respect to $x$.

The derivative of $\frac{1}{\sqrt{x}}$ as it pertains to the denominator.