Limits and Continuity

0

Undefined

9

6

Jump discontinuity

Point discontinuity

Removable discontinuity

Infinite discontinuity

Function $f$ has only one-sided limit existing near $x=a$ indicating a possible jump in continuity.

Only continuity could be concluded, while differentiability remains uncertain without further information.

Function $f$ is both continuous and differentiable at $x=a$.

There may be an essential discontinuity since limits are involved without specifying values.

Using partial fraction decomposition on each term individually.

None; jump discontinuities cannot be removed from an infinite series representation.

Multiplying each term by a decreasing sequence converging to zero.

Adding constants term-by-term until convergence at $x=c$.

Limits of a function as x approaches a particular value from one direction only, either from the left or from the right

Limits of a function as x approaches a particular value from the left only

Limits of a function as x approaches a particular value from both the left and the right

Limits of a function as x approaches a particular value from the right only

$g(k) = \lim_{x \to k} g(x)$

$\lim_{x \to k} g(x) = 0$

$g(k) = 0$

$g(k) = k$

Assume that the speed of convergence is larger than that of divergence.

Define $h(x)$ to be equal to the left-sided limit as $x$ approaches 1.

Add a constant term from the numerator to the denominator to achieve continuity.

Define $h(1)$ to be equal to its limit as $x$ approaches 1.

Take limits from only one side—left or right—to define $f(c)$.

Redefine $f(c)$ to equal $\lim_{x \to c} f(x)$.

Use integration around $x=c$ to fill in the undefined point.

Apply L'Hôpital's Rule until $f(c)$ becomes defined.

Differentiation and integration

Substitution and simplification

Approximation and estimation

Factoring and rationalization

$\frac{1}{L}$

$L$

$c$

$0$