Flashcard: LOGARITHMIC LIMITS AND DERIVATIVES — 16 carte

Derivative of product — rule?

$(uv)' = u'v + uv'$

Primitive of $rac{u'}{u}$ — function?

$oxed{ ext{Primitive} = ext{ln}|u| + C}$

Limit of $ ext{ln } x$ at zero?

$- ext{Infinity}$

Limit of $ ext{ln } x$ at infinity?

$+ ext{Infinity}$

Limit of $f(x)$ at zero or infinity?

Depends on the function's behavior near the point.

Limit of $ ext{ln } U(x)$ — when $U(x) o ext{?}$

$+ ext{Infinity}$ if $U(x) o+ ext{Infinity}$; $- ext{Infinity}$ if $U(x) o 0^+$; $ o ext{ln }k$ if $U(x) o k>0$.

Limit calculation techniques?

Use L'Hôpital, dominant terms, substitution, or algebraic simplification.

Limit of $x^eta ext{ln } x$ as $x o 0^+$?

$0$ for any real $eta$.

Limit of $rac{ ext{ln } x}{x^eta}$ as $x o ext{?}$

$0$ as $x o + ext{Infinity}$ for $eta>0$.

Behavior of $ ext{ln } x$ near zero?

Tends to $- ext{Infinity}$ as $x o 0^+$.

Behavior of $ ext{ln } x$ at infinity?

Tends to $+ ext{Infinity}$ as $x o + ext{Infinity}$.

Limit of $ ext{ln } U(x)$ when $U(x) o 0^+$?

$- ext{Infinity}$.

Limit of $ ext{ln } U(x)$ when $U(x) o + ext{Infinity}$?

$+ ext{Infinity}$.

Limit of $ ext{ln } U(x)$ when $U(x) o k>0$?

$ ext{ln }k$.

Key property of $ ext{ln } x$?

Diverges to $- ext{Infinity}$ at zero and $+ ext{Infinity}$ at infinity.

Why use the primitive of $rac{u'}{u}$?

To integrate functions involving ratios of derivatives and functions.