Common Limit Patterns

The following patterns are useful after checking the expression's domain and the direction of approach.

Expression	Limit
$\displaystyle\lim_{x\to\infty} e^x$	$\infty$
$\displaystyle\lim_{x\to-\infty} e^x$	$0$
$\displaystyle\lim_{x\to\infty}\ln x$	$\infty$
$\displaystyle\lim_{x\to0^+}\ln x$	$-\infty$
$\displaystyle\lim_{x\to\pm\infty}\frac{b}{x^r}$, $r>0$ where defined	$0$
$\displaystyle\lim_{x\to0}\frac{\sin x}{x}$	$1$
$\displaystyle\lim_{x\to0}\frac{1-\cos x}{x}$	$0$
$\displaystyle\lim_{x\to0}\frac{e^x-1}{x}$	$1$

For a polynomial

$$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0,$$

the leading term determines its end behavior. As $x\to\infty$, the sign is the sign of $a_n$. As $x\to-\infty$, an even degree preserves that sign and an odd degree reverses it.

For a rational function $p(x)/q(x)$, compare the polynomial degrees:

Degree comparison	Limit as $x\to\pm\infty$
$\deg p<\deg q$	$0$
$\deg p=\deg q$	ratio of leading coefficients
$\deg p>\deg q$	analyze leading powers; the value is generally unbounded or has no finite limit

These are patterns, not replacements for hypotheses. One-sided behavior and domain restrictions still matter.

Using the patterns

Trigonometric arguments must be in radians. For a constant $k\ne0$, set $u=kx$:

$$\lim_{x\to0}\frac{\sin(kx)}{x} =k\lim_{u\to0}\frac{\sin u}{u}=k.$$

For $k=0$, the quotient is identically zero for $x\ne0$. If angles are measured in degrees instead, the basic sine quotient has limit $\pi/180$, not $1$.

In the polynomial discussion, take $a_n\ne0$; growth to infinity requires $n\ge1$. A constant polynomial keeps its constant value. Degree comparisons assume nonzero polynomials $p,q$; a zero numerator gives zero wherever the quotient is defined. The negative-infinity power pattern applies only when the chosen real power is defined arbitrarily far to the left (not, for example, $x^{1/2}$).

My reading · Opened notes are remembered in this browser.