Limits and Continuity

Jump discontinuity

Point discontinuity

Removable discontinuity

Infinite discontinuity

0

Undefined

9

6

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

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

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

$g(k) = 0$

$g(k) = k$

It depends on the type of discontinuity

It depends on the value of the function at the point of discontinuity

Yes

No

4

Undefined

$\frac{8}{3}$

1

The limit as $p$ approaches zero from either side goes towards infinity, indicating an essential singularity.

There’s no need for cancellation since division by zero inherently creates non-removable discontinuities without exceptions.

The left-hand and right-hand limits of $h(p)$ as $p$ approaches zero are not equal, demonstrating a jump continuity.

The factor $p$ in both numerator and denominator can be cancelled out, leaving a polynomial.

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.