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Limit Properties and Rules Cheat Sheet, Cheat Sheet of Mathematics

Acharya Nagarjuna University Mathematics

A cheat sheet on limit properties and rules, including limit to a point, limit to infinity, indeterminate forms, common limits, and limit rules such as the squeeze theorem and l'hopital's rule.

Typology: Cheat Sheet

2017/2018

Uploaded on 09/13/2018

pukozivubi 🇮🇹

6 documents

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Symbolab Limits Cheat Sheet

Limit Properties:

If the limit of 𝑓(𝑥), and 𝑔(𝑥) exists, then the following apply:

• lim

𝑥→𝑎 𝑥 = 𝑎

• lim

𝑥→𝑎(𝑓(𝑥))𝑐= (lim

𝑥→𝑎 𝑓(𝑥))𝑐

• lim

𝑥→𝑎[𝑓(𝑥)± 𝑔(𝑥)] = lim

𝑥→𝑎 𝑓(𝑥)± lim

𝑥→𝑎 𝑔(𝑥)

• lim

𝑥→𝑎[𝑐 ⋅ 𝑓(𝑥)] = 𝑐 ⋅ lim

𝑥→𝑎[𝑓(𝑥)]

• lim

𝑥→𝑎[𝑓(𝑥)⋅ 𝑔(𝑥)] = lim

𝑥→𝑎 𝑓(𝑥)⋅ lim

𝑥→𝑎 𝑔(𝑥)

• lim

𝑥→𝑎 [𝑓(𝑥)

𝑔(𝑥)] = lim

𝑥→𝑎 𝑓(𝑥)

lim

𝑥→𝑎 𝑔(𝑥), where lim

𝑥→𝑎 𝑔(𝑥)≠ 0

Limit to Infinity Properties:

For lim

𝑥→𝑐 𝑓(𝑥)=

∞

,lim

𝑥→𝑐 𝑔(𝑥)= 𝐿 , the following applies:

• lim

𝑥→𝑐[𝑓(𝑥)± 𝑔(𝑥)] =

∞

• lim

𝑥→𝑐[𝑓(𝑥)⋅ 𝑔(𝑥)] =

∞

,𝐿 > 0

• lim

𝑥→𝑐[𝑓(𝑥)⋅ 𝑔(𝑥)] = −

∞

, 𝐿 < 0

• lim

𝑥→𝑐

𝑔(𝑥)

𝑓(𝑥)= 0

• lim

𝑥→

∞

𝑎𝑥𝑛=

∞

, 0 < 𝑎

• lim

𝑥→−

∞

𝑎𝑥𝑛=

∞

, 𝑛 is even,𝑎 > 0

• lim

𝑥→−

∞

𝑎𝑥𝑛= −

∞

, 𝑛 is odd, 𝑎 > 0

• lim

𝑥→

∞

𝑐

𝑥𝑎= 0

Discover Cheat Sheet of Mathematics Acharya Nagarjuna University

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Symbolab Limits Cheat Sheet

Limit Properties:

If the limit of 𝑓(𝑥), and 𝑔(𝑥) exists, then the following apply:

lim

𝑥→𝑎

lim

𝑥→𝑎

𝑐

= (lim

𝑥→𝑎

𝑐

lim

𝑥→𝑎

[

)]

= lim

𝑥→𝑎

± lim

𝑥→𝑎

lim

𝑥→𝑎

[𝑐 ⋅ 𝑓(𝑥)] = 𝑐 ⋅ lim

𝑥→𝑎

[𝑓(𝑥)]

lim

𝑥→𝑎

[𝑓(𝑥) ⋅ 𝑔(𝑥)] = lim

𝑥→𝑎

𝑓(𝑥) ⋅ lim

𝑥→𝑎

lim

𝑥→𝑎

[

𝑓

( 𝑥

)

𝑔

( 𝑥

)

] =

lim

𝑥→𝑎

𝑓

( 𝑥

)

lim

𝑥→𝑎

𝑔

( 𝑥

)

, where lim

𝑥→𝑎

Limit to Infinity Properties:

For lim

𝑥→𝑐

= ∞, lim

𝑥→𝑐

= 𝐿 , the following applies:

lim

𝑥→𝑐

[𝑓(𝑥) ± 𝑔(𝑥)] = ∞

lim

𝑥→𝑐

[

)]

lim

𝑥→𝑐

[𝑓(𝑥) ⋅ 𝑔(𝑥)] = − ∞, 𝐿 < 0

lim

𝑥→𝑐

𝑔(𝑥)

𝑓

( 𝑥

)

lim

𝑥→ ∞

𝑛

lim

𝑥→− ∞

𝑛

= ∞, 𝑛 is even, 𝑎 > 0

lim

𝑥→− ∞

𝑛

= − ∞, 𝑛 is odd, 𝑎 > 0

lim

𝑥→ ∞

𝑐

𝑥

𝑎

Indeterminate Forms:

0

∞

0

∞

0

0 ⋅ ∞
∞ − ∞

Common Limits:

lim

𝑥→ ∞

𝑘

𝑥

𝑘

lim

𝑥→ ∞

𝑥

𝑥+𝑘

𝑥

−𝑘

lim

𝑥→ 0

1

𝑥 = 𝑒

Limit Rules:

Limit of a Constant: lim

𝑥→𝑎

Basic Limit: lim

𝑥→𝑎

Squeeze Theorem: Let 𝑓, 𝑔 and ℎ be functions such that for all 𝑥 ∈ [𝑎, 𝑏]

(except possible at the limit point c), 𝑓(𝑥) ≤ ℎ(𝑥) ≤ 𝑔(𝑥). Also suppse that

lim

𝑥→𝑐

𝑓(𝑥) = lim

𝑥→𝑐

𝑔(𝑥) = 𝐿, then for any 𝑐, 𝑎 ≤ 𝑐 ≤ 𝑏, lim

𝑥→𝑐

L’Hopital’s Rule : For lim

𝑥→𝑎

𝑓(𝑥)

𝑔

( 𝑥

)

, if lim

𝑥→𝑎

𝑓(𝑥)

𝑔

( 𝑥

)

0

or lim

𝑥→𝑎

𝑓(𝑥)

𝑔

( 𝑥

)

± ∞

, then

lim

𝑥→𝑎

𝑓(𝑥)

𝑔

( 𝑥

)

= lim

𝑥→𝑎

𝑓

′(𝑥)

𝑔

′(𝑥)

Divergence Criterion: If there exists two sequences, {𝑥

𝑛

𝑛= 1

∞

and {𝑦

𝑛

𝑛= 1

∞

with: 𝑥

𝑛

≠ 𝑐 and lim

𝑛→ ∞

𝑛

= lim

𝑛→ ∞

𝑛

lim

𝑛→ ∞

𝑛

) ≠ lim

𝑛→ ∞

𝑛

), then lim

𝑥→𝑐

𝑓(𝑥) does not exist

Limit Chain Rule: If lim

𝑢→𝑏

= 𝐿, and lim

𝑥→𝑎

= 𝑏, and 𝑓

is continuous at

𝑥 = 𝑏, Then: lim

𝑥→𝑎

Limit Properties and Rules Cheat Sheet, Cheat Sheet of Mathematics

Related documents

Partial preview of the text

Download Limit Properties and Rules Cheat Sheet and more Cheat Sheet Mathematics in PDF only on Docsity!

Symbolab Limits Cheat Sheet

[

)]

[𝑓(𝑥)]

[

] =

= ∞, lim

[𝑓(𝑥) ± 𝑔(𝑥)] = ∞

[

)]

[𝑓(𝑥) ⋅ 𝑔(𝑥)] = − ∞, 𝐿 < 0

= ∞, 𝑛 is even, 𝑎 > 0

= − ∞, 𝑛 is odd, 𝑎 > 0