Limits Form 2 (Infinity by Infinity)

1. Statement of the Concept (∞ / ∞ Form)

When both the numerator and the denominator of a function tend to infinity as [x → a] or [x → ∞], the limit is said to be of the indeterminate form:

[∞ / ∞]

Such limits cannot be evaluated directly and require algebraic simplification.

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2. Clear Explanation and Mathematical Understanding

Why is [∞ / ∞] indeterminate?

• Infinity is not a number
• Different functions grow at different rates
• The ratio depends on which grows faster

Example:

• [\dfrac{x}{x} → 1]
• [\dfrac{x}{x^{2}} → 0]
• [\dfrac{x^{2}}{x} → ∞]

So, form alone does not decide the value.

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Core Principle (Most Important for Exams)

In limits of type [∞ / ∞], divide every term by the highest power of [x] present in the denominator (or numerator).

This converts the expression into a finite form.

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3. Key Features of (∞ / ∞) Limits

• Always an indeterminate form
• Evaluated using algebraic division
• Highest power of [x] decides the limit
• Widely asked in board exams & competitive exams

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4. Explaining (∞ / ∞) Form Using the Given Example

[
\lim_{x \to \infty} \dfrac{ax^{2} + bx + c}{dx^{2} + ex + f}
]

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Step 1: Identify the Form

As [x → ∞]:

• Numerator: [ax² + bx + c → ∞]
• Denominator: [dx² + ex + f → ∞]

So, the limit is of the indeterminate form:

[∞ / ∞]

This means we cannot substitute x = ∞ directly.

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Step 2: Why Is This Form Indeterminate?

Students often think:

“∞ divided by ∞ should be 1”

This is wrong.

Reason:

• Different terms grow at different speeds
• The result depends on which term dominates

So we must simplify algebraically.

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Step 3: Identify the Highest Power of [x]

• Highest power of [x] in numerator = [x²]
• Highest power of [x] in denominator = [x²]

This step decides the entire method.

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Step 4: Divide Every Term by the Highest Power [x²]

Divide numerator and denominator by [x²]:

[\dfrac{\dfrac{ax^{2}}{x^{2}} + \dfrac{bx}{x^{2}} + \dfrac{c}{x^{2}}}{\dfrac{dx^{2}}{x^{2}} + \dfrac{ex}{x^{2}} + \dfrac{f}{x^{2}}}]

Simplifying each term:

[= \dfrac{a + \dfrac{b}{x} + \dfrac{c}{x^{2}}}{d + \dfrac{e}{x} + \dfrac{f}{x^{2}}}]

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Step 5: Apply the Limit [x → ∞]

As [x → ∞]:

• [\dfrac{b}{x} → 0]
• [\dfrac{c}{x^{2}} → 0]
• [\dfrac{e}{x} → 0]
• [\dfrac{f}{x^{2}} → 0]

So the expression becomes:

[
\dfrac{a + 0 + 0}{d + 0 + 0}
]

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Step 6: Final Result

[
\boxed{\lim_{x \to \infty} \dfrac{ax^{2} + bx + c}{dx^{2} + ex + f} = \dfrac{a}{d}}
]

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5. Key Conclusion (Must Remember for Exams)

When the highest powers of [x] in numerator and denominator are equal,
the limit is equal to the ratio of their coefficients.

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6. Important Results to Remember

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7. Conceptual Questions with Solutions

1. Why is [∞ / ∞] called an indeterminate form?

Although both numerator and denominator become infinite, their ratio is not fixed.
Different growth rates produce different results. Hence, the value cannot be determined without simplification.

2. How do we remove the [∞ / ∞] form?

By dividing every term in numerator and denominator by the highest power of [x].
This converts infinite terms into finite or zero terms.

3. If degrees of numerator and denominator are equal, what is the limit?

The limit equals the ratio of coefficients of the highest power terms.

4. If denominator has a higher degree than numerator, what happens?

The denominator grows faster, so the limit becomes zero.

5. If numerator has higher degree, why does the limit become infinite?

Because the numerator dominates the denominator, making the fraction grow without bound.

6. Can we cancel infinity directly?

 No.
Infinity is not a number. Only algebraic simplification is valid.

7. Why do lower power terms vanish after division?

Because terms like [1/x], [1/x^{2}] tend to zero as [x → ∞].

8. Is factorisation needed in all [∞ / ∞] limits?

 No.
Division by highest power is usually sufficient.

9. Are these limits important for continuity?

Yes.
They are used to study end behavior and asymptotes.

10. Are these questions asked directly in board exams?

Very frequently — often as 1, 2, or 3 mark questions.

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8. FAQ / Common Misconceptions

1. Can [∞ / ∞] ever be equal to 1?

Yes, if numerator and denominator grow at the same rate.

2. Does [∞ / ∞] always mean infinity?

No. It may be 0, finite, or infinite.

3. Can I apply L’Hôpital’s Rule here?

Not at beginner level.
Algebraic method is expected in school exams.

4. Should I substitute infinity directly?

Never. Always simplify first.

5. Do constants matter in these limits?

Only coefficients of highest powers matter.

6. Can negative infinity occur?

Yes, depending on signs of leading coefficients.

7. Is this topic linked with graphs?

Yes, it explains end behavior of graphs.

8. Why is division by highest power preferred?

Because it normalizes growth and removes infinity.

9. Are these limits continuous everywhere?

Continuity depends on function definition, not just limit.

10. Is this topic required for calculus?

Yes, it is foundational.

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8. Practice Questions with Step-by-Step Solutions

Question 1. Evaluate:
[\lim_{x → ∞} \dfrac{3x + 5}{2x − 1}]

Step-by-Step Solution:

Highest power of [x] is [x¹]

Divide numerator and denominator by [x]

[\dfrac{3 + 5/x}{2 − 1/x}]

As [x → ∞], [5/x → 0] and [1/x → 0]

Limit becomes:

[\dfrac{3}{2}]

Final Answer:
[\dfrac{3}{2}]

Question 2. Evaluate:
[\lim_{x → ∞} \dfrac{4x^{2} − x}{x^{2} + 3}]

Step-by-Step Solution:

Highest power = [x²]

Divide all terms by [x²]

[\dfrac{4 − 1/x}{1 + 3/x²}]

As [x → ∞], fractions → 0

Limit:

[4]

Final Answer:
[4]

Question 3. Evaluate:
[\lim_{x → ∞} \dfrac{5x}{x^{2} + 1}]

Step-by-Step Solution:

Highest power = [x²]

Divide by [x²]

[\dfrac{5/x}{1 + 1/x²}]

As [x → ∞], [5/x → 0]

Final Answer:
[0]

Question 4. Evaluate:
[\lim_{x → ∞} \dfrac{2x^{3} + x}{x^{3} − 4}]

Step-by-Step Solution:

Divide by [x³]

[\dfrac{2 + 1/x²}{1 − 4/x³}]

As [x → ∞], fractions vanish

Final Answer:
[2]

Question 5. Evaluate:
[\lim_{x → ∞} \dfrac{x^{2} + 1}{x}]

Step-by-Step Solution:

Divide by [x]

[x + 1/x]

As [x → ∞], expression → ∞

Final Answer:
[∞]