# Infinite Limits in Calculus

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Infinite Limits in Calculus

**Introduction:**

There is no real number represented by the symbol for infinity. When a domain and range of a function exceeds all finite boundaries, we use to explain how the function behaves. A limit is a value that an expression converges to as one or more variables get closer to a certain value in mathematics. In calculus and statistics, limits are important. For the purpose of representing the behavior of functions whose values could reach large magnitudes, infinite limits provide some symbols and expressions. There are two types of limits one is positive limit and second one is minus infinity. Here we defined the infinite limits as positive limits and minus limits.

**Definition:**

**Positive Infinite Limits:**

As x gets closer to infinity, we state that (x) has the limit M and write,

Lim x→ ∞ f(x) = M

If there is a number N that corresponding to every number δ, then for any x

X > N this implies that |f(x) – M| < δ

**Minus Infinite Limits:**

As x gets closer to minus infinity, we state that (x) has the limit M and write,

Lim x→ -∞ f(x) = M

If there is a number N that corresponding to every number** δ**, then for any x

X < N this implies that |f(x) – M| < δ

**Important Note:**

Lim x→ ∞ f(x) = M if x gets arbitrarily close to M as it moves in a positive direction away from the origin. Similar results apply on Lim x→ -∞ f(x) = M if x goes arbitrarily close to M as it moves away from the origin in a negative direction.

**Examples:**

**Example No.1:**

**Given that**

Lim x→ ∞ (1 / x) =?

**Solution:**

As we know that by the definition of infinite li limits,

Lim x→ ∞ f(x) = Lim x→ ∞ (1 / x)

By apply the limit ∞ on 1 / x

It gives,

Lim x→ ∞ (1 / x) = 1 / ∞ = 0

Hence,

Lim x→ ∞ (1 / x) = 0

**Example No.2:**

**Given that**

Lim x→ ∞ (x -5)

**Solution:**

Lim x→ ∞ f(x) = Lim x→ ∞ (x -5)

By apply the Lim x→ ∞ on (x -5)

Lim x→ ∞ (x) - Lim x→ ∞ (5) = ∞ - 5

By the subtraction property of infinity it gives,

⇒ ∞

Hence,

Lim x→ ∞ (x -5) = ∞

**Example No.3:**

Given That:

Lim x→ ∞ (x^2+2x) =?

**Solution:**

Lim x→ ∞ f(x) = Lim x→ ∞ (x^2+2x)

By apply the Lim x→ ∞ on (x^2+2x)

Lim x→ ∞ (x^2) + Lim x→ ∞ (2x) = ∞^2 + 2 Lim x→ ∞ (x)

⇒ ∞ + 2(∞)

By the addition property of infinity it gives,

⇒ ∞

Hence,

Lim x→ ∞ (x^2+2x) = ∞

**Example No.4:**

**Given That:**

Find limit Lim x→ ∞ (e^1/x)

**Solution:**

Lim x→ ∞ f(x) = Lim x→ ∞ (e^1 / x)

By apply the limit ∞ on e^1 / x

It gives,

Lim x→ ∞ (e^1 / x) = e^1 / ∞ = e^0

Since,

e^0 = 1

Hence,

Lim x→ ∞ (e^1 / x) = 1

**Properties of Infinite Limits:**

**Laws of Infinite Limits:**

There are some laws of infinite limit like,

- Sum law
- Subtraction law
- Constant multiplication law
- Multiplication law
- Quitrents law
- Power law
- Root law

If M and N are real numbers and

lim (x→∞) f(x) = M

And,

lim (x→∞) g(x)= N

Then

**Sum Law:**

Lim (x→∞) (f(x) + g(x)) = M + N

**Subtraction Law:**

Lim (x→∞) (f(x) - g(x)) = M- N

**Constant Multiple Law:**

Lim (x→∞) (k. f(x)) = k. M

**Product Rule:**

Lim (x→ ∞) (f(x). g(x)) = M. N

**Quotient Rule:**

Lim (x→∞) f(x) / g(x) = M / N Where N is not equal to zero.

**Power Law:**

Lim (x→∞) [(f(x))] ^n= M^n Where is a positive integer.

**Root Law:**

Lim (x→∞) [(f(x))] ^ (1⁄n) = M^ (1⁄n) Where is a positive integer.

**FAQs**

**1. What is meant by infinite limits in calculus?**

As x gets closer to infinity, we state that (x) has the limit M and write,

Lim x→ ∞ f(x) = M

If there is a number N that corresponding to every number δ, then for any x

X > N this implies that |f(x) – M| < δ

**2. Write down the laws of infinite limits?**

There are some laws of infinite limit like,

- Sum law
- Subtraction law
- Constant multiplication law
- Multiplication law
- Quitrents law
- Power law
- Root law

**3. What will be the answer if infinite limit is applied on constant function?**

When infinite limit is applied on constant function the answer will be the same as constant.

**4. Are there boundaries at infinity?**

Since infinity cannot have a specific value, we conclude that the infinite limit is meaningless.

**5. Give one example of infinite limit?**

**Given That:**

Find limit Lim x→ ∞ (e^1/x)

**Solution:**

Lim x→ ∞ f(x) = Lim x→ ∞ (e^1 / x)

By apply the limit ∞ on e^1 / x

It gives,

Lim x→ ∞ (e^1 / x) = e^1 / ∞ = e^0

Since,

e^0 = 1

Hence,

Lim x→ ∞ (e^1 / x) = 1

**References:**

Thomas, G.B., Weir, M.D., Hass, J., Heil, C. and Behn, A., 2010. *Thomas' calculus: Early transcendentals* (p. 510). Boston: Pearson.

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*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

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