One-sided limits — left-hand, right-hand: Question bank

Pictures to keep in mind

The two figures below are the mental images every item on this page leans on: a number-line "approach from both sides" picture, and the $ε$ – $δ$ band picture that shows why one inequality secretly carries two sides.

Figure — One-sided limits — left-hand, right-hand

Look at the second figure: the horizontal band of half-width $ε$ around $L$ is what we want $f (x)$ to land in; the vertical strip of half-width $δ$ around $a$ is where $x$ is allowed to roam. That strip straddles $a$ — points to its left and to its right are both inside $0 < ∣ x - a ∣ < δ$ . That is the geometric reason a single inequality demands agreement from both sides.

True or false — justify

For a claimed one-sided limit $lim_{x \to a^{+}} f (x) = L$ : " $+$ " means we approach $a$ using only $x > a$ .

True or false:

x \to 5^{-}

means

x

takes negative values.

False. It means

x

approaches

5

through numbers smaller than

5

(like

4.9, 4.99

), which are all positive; the minus is a direction, not a sign.

True or false: If

lim_{x \to a^{-}} f = lim_{x \to a^{+}} f

, the two-sided limit exists and equals that common value.

True. Equal one-sided limits are exactly the condition the boxed rule requires; the shared value is the two-sided limit.

True or false: If both one-sided limits exist, the two-sided limit must exist.

False. They must also be equal. For

∣ x ∣/ x

0

both sides exist (

+ 1

and

- 1

) but disagree, so no two-sided limit.

True or false: The value

f (a)

is needed to compute

lim_{x \to a^{+}} f (x)

False. A limit only reads the neighbourhood; the point

a

itself is never plugged in and can even be undefined.

True or false:

lim_{x \to 0^{+}} \frac{1}{x} = + \infty

is an existing (real-valued) limit.

False.

+ \infty

is not a real number; we say the limit diverges to

+ \infty

— it names a behaviour, not an existing limit.

True or false: If

lim_{x \to a^{-}} f = + \infty

and

lim_{x \to a^{+}} f = + \infty

, then

lim_{x \to a} f = + \infty

True in the sense that both sides "agree" on diverging to

+ \infty

(like

1/ x^{2}

0

); we write the two-sided divergence as

+ \infty

, though it is still a non-existent finite limit.

True or false: A continuous function can have unequal one-sided limits at a point in its domain.

False. Continuity at

a

demands both one-sided limits equal

f (a)

; unequal sides are exactly what breaks continuity (a jump).

True or false: If

f

is defined only for

x \geq a

, then

lim_{x \to a} f (x)

can still exist two-sided.

False. With no domain on the left the left-hand limit is undefined, so the two-sided limit cannot exist; only the right-hand limit is meaningful.

True or false:

lim_{x \to 3^{-}} ⌊ x ⌋ = ⌊ 3 ⌋ = 3

False. Just below

3

, values like

2.999

floor to

2

, so the left-hand limit is

2

, not

3

; the floor jumps at the integer.

Spot the error

Each line states a flawed piece of reasoning. Name what went wrong.

g (2) = 7

, and

g

is defined there, so

lim_{x \to 2} g (x) = 7

." (piecewise

g

from the parent note)

Error: the limit ignores

g (2)

. The one-sided rules

x + 1

and

3 x - 3

both approach

3

, so the limit is

3

; the point value

g (2) = 7

is irrelevant.

∣ x ∣/ x

0

: since

∣ x ∣ = x

always,

f (x) = 1

on both sides, so the limit is

1

Error:

∣ x ∣ = x

only for

x > 0

. For

x < 0

∣ x ∣ = - x

, giving

f (x) = - 1

, so the sides disagree and no two-sided limit exists.

"Both sides of

1/ x

0

blow up, so the two-sided limit is

\infty

Error: the sides go to opposite infinities (

+ \infty

from the right,

- \infty

from the left), so they disagree; the limit does not exist and cannot be labelled a single

\infty

x

0

: left-hand limit is

- 0.001

which is imaginary, so the limit is imaginary."

Error: over the reals

x

has no left domain at all, so the left-hand limit is undefined (not imaginary); only

lim_{x \to 0^{+}} x = 0

exists.

x \to a^{-}

picks the smaller root/branch of

f

Error: the minus is purely a direction of approach (

x < a

); it says nothing about which branch or value of

f

to choose.

"Since

lim_{x \to 3^{+}} ⌊ x ⌋ = 3 = ⌊ 3 ⌋

, the floor is right-continuous and hence continuous at

3

Error: right-continuity alone is not continuity. The left-hand limit is

2 \neq = 3

, so

⌊ x ⌋

is discontinuous at

3

despite matching from the right.

Why questions

Why must a two-sided limit require the sides not just to exist but to be equal?

A limit is a promise of a single destination

L

reachable from every approach; if left heads to

2

and right to

5

, no single

L

can honour the promise, so it fails.

Why does the

ε

–

δ

condition

0 < ∣ x - a ∣ < δ

automatically encode "both sides"?

Picture the

δ

-strip around

a

(second figure): it extends

δ

to the left of

a

and

δ

to the right, so

∣ x - a ∣ < δ

covers

- δ < x - a < 0

and

0 < x - a < δ

; the two-sided limit is the logical AND of these two halves.

Why is

f (a)

deliberately excluded (via

0 < ∣ x - a ∣

, not

0 \leq

The strict

0 < ∣ x - a ∣

punches a hole exactly at

x = a

, so the limit describes where

f

is heading near

a

and stays meaningful even when

f (a)

is undefined or an outlier.

Why can a limit be written "

= + \infty

" yet still be called non-existent?

The symbol

+ \infty

faithfully records that outputs grow past every bound, but since

+ \infty

is not a real number, there is no real value

L

to equal — so no finite limit exists; "diverges to

+ \infty

" is the honest phrasing.

Why does the floor function fail to have a two-sided limit at every integer but succeed everywhere else?

At an integer the floor value jumps by

1

, so left and right limits differ; strictly between integers

⌊ x ⌋

is constant, so both sides agree and the limit exists.

Why does approaching a vertical asymptote from the left of

1/ x

give

- \infty

but the right give

+ \infty

Tiny negative denominators make

1/ x

hugely negative, tiny positive ones make it hugely positive; the sign of the shrinking denominator flips the output's sign.

Edge cases

f (x) = x

x = 0

: which one-sided limit is even meaningful, and what is the two-sided verdict?

Only the right-hand limit is meaningful,

lim_{x \to 0^{+}} x = 0

; the left has no real domain, so the two-sided limit does not exist.

A function defined only on

[a, b]

: what is the status of

lim_{x \to a} f

and

lim_{x \to b} f

Only

lim_{x \to a^{+}} f

and

lim_{x \to b^{-}} f

can exist (approach from inside the domain); the two-sided limits at the endpoints do not exist since one side has no points.

f (x) = 1/ x^{2}

x = 0

: do the sides agree, and does the two-sided limit "exist"?

Both sides diverge to

+ \infty

, so they agree in behaviour and we write

lim_{x \to 0} 1/ x^{2} = + \infty

; it is still a non-existent finite limit but a genuine two-sided divergence.

A removable discontinuity where both sides equal

4

but

f (a) = 9

: does

lim_{x \to a} f

exist?

Yes, the two-sided limit is

4

because both sides agree; the mismatched

f (a) = 9

only means

f

is discontinuous, not that the limit fails.

f (x) = sin (1/ x)

x \to 0^{+}

: does the right-hand limit exist?

No — near

0

the input

1/ x

races through infinitely many cycles, so

f

oscillates between

- 1

and

1

without settling; no single approach value, hence no one-sided limit.

An isolated point of the domain (say

f

defined at

a

but nowhere in a punctured neighbourhood): can any one-sided limit exist?

No. With no domain points approaching

a

from either side, neither one-sided nor two-sided limit is defined; only

f (a)

as a plain value exists.

Both one-sided limits equal

2

and

f (a) = 2

as well — is

f

discontinuous, and is there any jump here?

No to both. All three agree, so

f

is continuous at

a

; there is no jump — a jump discontinuity requires the two one-sided limits to differ, which does not happen here.

Connections

One-sided Limits — Left-hand & Right-hand
Limit of a function — intuitive & ε-δ definition
Continuity at a point
Jump, removable & infinite discontinuities
Vertical asymptotes
Greatest integer / floor function
Differentiability — left & right derivatives