Limits Form 3: Trigonometrical Limits

1. Introduction

Trigonometrical limits are those limits in which trigonometric functions such as [sin x], [cos x], [tan x] appear, and the value of the limit is evaluated using standard trigonometric results.

These limits are mostly evaluated when:

• [x → 0]
• The expression gives 0/0 form

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2. Fundamental Idea Behind Trigonometrical Limits

When [x → 0]:

• $ \displaystyle \sin x\to 0$
• $\displaystyle \tan x\to 0$
• $ \displaystyle \left( {1-\cos x} \right)\to 0$

So, many trigonometric limits give Zero by Zero (0/0) form, which is indeterminate.

To solve such limits, we use standard trigonometric results, not direct substitution.

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3. Important Standard Trigonometrical Limits

(Heart of this topic – Must Memorise)

(All angles must be in radians)

1.  [\lim_{x → 0} \dfrac{\sin x}{x} = 1]

2.  [\lim_{x → 0} \dfrac{\tan x}{x} = 1]

3.  [\lim_{x → 0} \dfrac{1 − \cos x}{x^{2}} = \dfrac{1}{2}]

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4. Derived Standard Results (Using Above Limits)

Using basic identities:

• [\lim_{x → 0} \dfrac{x}{\sin x} = 1]
• [\lim_{x → 0} \dfrac{x}{\tan x} = 1]
• [\lim_{x → 0} \dfrac{\sin ax}{ax} = 1]
• [\lim_{x → 0} \dfrac{\tan ax}{ax} = 1]

These results are very frequently used in exams.

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5. Trigonometrical Limits – Important General Forms & Inverse Trig Limits

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A. Trigonometric Limits Using General Forms

[
\lim_{x \to 0} \dfrac{\tan (m x)}{n x}
]

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Step 1: Identify the Nature of the Limit

As [x → 0]:

• [\tan(mx) → 0]
• [nx → 0]

So, the form is:

[0 / 0] → Indeterminate Form

Hence, standard trigonometric limits must be applied.

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Step 2: Convert Expression into Standard Form

We know the standard result:

[\lim_{θ → 0} \dfrac{\tan θ}{θ} = 1]

So we try to create this form.

Rewrite the given expression:

[\dfrac{\tan(mx)}{nx} = \dfrac{m}{n} \cdot \dfrac{\tan(mx)}{mx}]

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Step 3: Apply the Limit

As [x → 0]:

• [mx → 0]
• So,

[\dfrac{\tan(mx)}{mx} → 1]

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Step 4: Final Answer

[\boxed{\lim_{x \to 0} \dfrac{\tan(mx)}{nx} = \dfrac{m}{n}}]

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Key Exam Rule (Very Important)

Whenever you see
[\dfrac{\tan(ax)}{bx}] as [x → 0],
answer = [a / b]

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B. Other Very Important General Trig Forms

1. [\lim_{x \to 0} \dfrac{\sin(mx)}{nx}]

Solution Idea:

[
\dfrac{m}{n} \cdot \dfrac{\sin(mx)}{mx}
]

Final Answer:

[
\boxed{\dfrac{m}{n}}
]

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2. [\lim_{x \to 0} \dfrac{1 – \cos(mx)}{x^{2}}]

Standard Result Used:

[\lim_{θ → 0} \dfrac{1 – \cos θ}{θ^{2}} = \dfrac{1}{2}]

Final Answer:

[
\boxed{\dfrac{m^{2}}{2}}
]

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3. [\lim_{x \to 0} \dfrac{\sin(mx)}{\tan(nx)}]

Rewrite as:

[\dfrac{m}{n} \cdot \dfrac{\sin(mx)}{mx} \cdot \dfrac{nx}{\tan(nx)}]

Final Answer:

[
\boxed{\dfrac{m}{n}}
]

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C. Inverse Trigonometrical Function Limits

(Very Important)

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1. Fundamental Inverse Trig Limit

[
\lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1
]

Why?

As [x → 0]:

• [\sin⁻¹x ≈ x]
• So ratio → 1

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6. Other Standard Inverse Trig Limits

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D. Example-Based Explanation (Inverse Trig)

Example 1:

[
\lim_{x \to 0} \dfrac{\sin^{-1}(3x)}{x}
]

Step-by-Step Solution:

1. Multiply and divide by 3:

[
3 \cdot \dfrac{\sin^{-1}(3x)}{3x}
]

2. As [x → 0], [3x → 0]
3. Use standard limit:

[\dfrac{\sin^{-1} t}{t} → 1]

Final Answer:

[
\boxed{3}
]

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Example 2:

[
\lim_{x \to 0} \dfrac{\tan^{-1}(5x)}{2x}
]

Rewrite:

[
\dfrac{5}{2} \cdot \dfrac{\tan^{-1}(5x)}{5x}
]

Final Answer:

[
\boxed{\dfrac{5}{2}}
]

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E. Common Confusions (Clarified Clearly)

• Do not substitute x = 0 directly
• Do not cancel x inside trig functions
• Always try to convert to standard form
• Inverse trig limits behave like simple linear functions near zero

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F. Combined Trig + Inverse Trig Example

[\lim_{x \to 0}\dfrac{\sin^{-1}(2x) + \tan^{-1}(3x)}{x}]

Split the limit:

[\dfrac{\sin^{-1}(2x)}{x} + \dfrac{\tan^{-1}(3x)}{x}]

Apply standard results:

[
= 2 + 3 = \boxed{5}
]

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Final Takeaway for Students

Near zero:

• [\sin x ≈ x]
• [\tan x ≈ x]
• [\sin⁻¹x ≈ x]
• [\tan⁻¹x ≈ x]

This single idea solves 90% of trig limits.

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6. Key Features of Trigonometrical Limits

• Mostly evaluated at [x → 0]
• Always involve standard results
• Angle must be in radians
• Cannot be solved by substitution
• Highly scoring and predictable questions

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7. Conceptual Questions with Solutions

1. Why do we use radians and not degrees in trigonometric limits?

Standard trigonometric limits are true only when angles are measured in radians.
If degrees are used, limits like [\sin x / x] do not approach 1.
Hence, radians are compulsory.

2. Why does [sin x / x] tend to 1 as [x → 0]?

As [x → 0], [sin x] and [x] become nearly equal in radians.
Their ratio therefore approaches 1.

3. Can we directly substitute [x = 0] in trigonometric limits?

No.
Direct substitution gives [0/0], which is indeterminate.
We must use standard results.

4. Why is [1 − cos x] divided by [x²] and not by [x]?

Because $\displaystyle \left( {1-\cos x} \right)=2{{\sin }^{2}}\left( {\frac{x}{2}} \right)$ and it behaves like [x²] near zero, not like [x].
That is why the correct standard form is
[\dfrac{1 − cos x}{x²}].

5. What happens if the angle is not in radians?

The standard results fail.
The limit value becomes incorrect.

6. Is [\tan x / x] always equal to 1?

Only when [x → 0].
For other values, it is not equal to 1.

7. Why are trigonometric limits important for calculus?

They are used in:

Differentiation

Continuity

Series expansion

8. Are these limits asked directly in exams?

Yes.
Often as direct questions or as part of larger problems.

9. Can we use identities to simplify trig limits?

Yes.
Identities are often used to convert expressions into standard forms.

10. Are trigonometric limits always of 0/0 type?

Mostly yes, but not always.
Some may reduce to finite values after simplification.

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8. FAQ / Common Misconceptions

1. [sin 0 / 0] = 1?

Wrong.
It is undefined. The limit equals 1, not the value.

2. Can I cancel x in [sin x / x]?

No.
They are not algebraic terms.

3. Can I apply algebraic limits directly?

Only after converting into standard trig forms.

4. Is [\sin x ≈ x] always true?

Only when [x → 0] and in radians.

5. Can we apply L’Hôpital’s Rule?

Not expected at school level.

6. Why does [(1 − \cos x)] go to zero faster?

Because it behaves like [x²] near zero.

7. Are these limits used in physics?

Yes, very frequently in small-angle approximations.

8. Is memorisation enough?

No. Understanding the application is necessary.

9. Can standard limits be modified?

Yes, by multiplying and dividing appropriately.

10. Is this topic compulsory?

Absolutely.

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9. Practice Questions with Step-by-Step Solutions

Question 1. Evaluate:
[\lim_{x → 0} \dfrac{\sin x}{x}]

Step-by-Step Solution:

As [x → 0], numerator → 0 and denominator → 0

Form is [0/0]

Using standard result:

[\lim_{x → 0} \dfrac{\sin x}{x} = 1]

Final Answer:
[1]

Question 2. Evaluate:
[\lim_{x → 0} \dfrac{\tan x}{x}]

Step-by-Step Solution:

As [x → 0], expression gives [0/0]

Using standard result:

[\lim_{x → 0} \dfrac{\tan x}{x} = 1]

Final Answer:
[1]

Question 3. Evaluate:
[\lim_{x → 0} \dfrac{\sin 3x}{x}]

Step-by-Step Solution:

Multiply and divide by 3:

[3 \dfrac{\sin 3x}{3x}]

As [x → 0], [3x → 0]

Using standard result:

[3 × 1 = 3]

Final Answer:
[3]

Question 4. Evaluate:
[\lim_{x → 0} \dfrac{1 − \cos x}{x^{2}}]

Step-by-Step Solution:

Form is [0/0]

Use standard result:

$ \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\dfrac{{1-\cos x}}{{{{x}^{2}}}}$ $=\underset{{x\to 0}}{\mathop{{\lim }}}\,2{{\sin }^{2}}\left( {\dfrac{x}{2}} \right)$ $=\dfrac{1}{2}$

Final Answer:
[\dfrac{1}{2}]

Question 5. Evaluate:
[\lim_{x → 0} \dfrac{x}{\sin x}]

Step-by-Step Solution:

Take reciprocal of standard result:

[\lim_{x → 0} \dfrac{x}{\sin x} = 1]

Final Answer:
[1]