Logarithmic Limits

1. Concept Overview: What Are Logarithmic Limits?

A logarithmic limit is a limit in which the given expression involves a logarithmic function, usually of the form:

• [\log(1 + x)]
• [\ln(1 + x)]
• [\log(1 + ax)]

These limits are generally evaluated when the argument of the logarithm approaches 1, because:

Logarithmic functions behave very simply near 1

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Why “1” Is Important in Logarithmic Limits?

Recall:

• [\ln 1 = 0]
• [\log 1 = 0]

So, whenever the expression inside the logarithm approaches 1, the numerator approaches 0, and the limit often becomes an indeterminate form [\dfrac{0}{0}].

That is exactly where limits are required.

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2. Base of Logarithm – A Crucial Clarification

In calculus and limits:

• Natural logarithm [\ln] (base e) is preferred
• Other logarithms are converted using:

[\log a = \dfrac{\ln a}{\ln 10}]

So first, we derive standard results using [\ln], and then extend them to other bases.

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3. First Standard Logarithmic Limit

Standard Result 1

[\lim_{x \to 0} \dfrac{\ln(1+x)}{x} = 1]

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Step-by-Step Derivation

Step 1: Observe the form

As [x → 0]:

• [\ln(1+x) → \ln 1 = 0]
• [x → 0]

So the limit is of the form:
[\dfrac{0}{0}] → Indeterminate

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Step 2: Use a known exponential limit

We already know the standard exponential limit:

[\lim_{x \to 0} (1 + x)^{\dfrac{1}{x}} = e]

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Step 3: Take natural logarithm on both sides

[\ln \left( (1 + x)^{\dfrac{1}{x}} \right) = \ln e]

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Step 4: Use logarithmic laws

Using [\ln a^b = b \ln a]:

[\dfrac{1}{x} \ln(1 + x) = 1]

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Step 5: Rearranging

[\dfrac{\ln(1+x)}{x} = 1]

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Step 6: Take limit

[\lim_{x \to 0} \dfrac{\ln(1+x)}{x} = 1]

✔ Derived, not memorized

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4. Second Standard Logarithmic Limit (With Coefficient)

Standard Result 2

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{x} = a]

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Step-by-Step Derivation

Step 1: Rewrite the expression

[\dfrac{\ln(1+ax)}{x} = a \cdot \dfrac{\ln(1+ax)}{ax}]

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Step 2: Take limit

[\lim_{x \to 0} a \cdot \dfrac{\ln(1+ax)}{ax}]

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Step 3: Apply Standard Result 1

Since [ax → 0] as [x → 0]:

[\lim_{ax \to 0} \dfrac{\ln(1+ax)}{ax} = 1]

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Step 4: Final Answer

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{x} = a]

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5. Logarithmic Limit with a Constant in Denominator

Standard Result 3

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{bx} = \dfrac{a}{b}]

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Step-by-Step Explanation

Step 1: Separate constants

[\dfrac{\ln(1+ax)}{bx} = \dfrac{1}{b} \cdot \dfrac{\ln(1+ax)}{x}]

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Step 2: Use Standard Result 2

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{x} = a]

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Step 3: Final Answer

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{bx} = \dfrac{a}{b}]

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6. Logarithmic Limit with Trigonometric Function

Standard Result 4

[\lim_{x \to 0} \dfrac{\ln(1+\sin x)}{x} = 1]

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Step-by-Step Derivation

Step 1: Multiply and divide by [\sin x]

[\dfrac{\ln(1+\sin x)}{x}
= \dfrac{\ln(1+\sin x)}{\sin x} \cdot \dfrac{\sin x}{x}]

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Step 2: Evaluate each limit separately

• [\lim_{x \to 0} \dfrac{\ln(1+\sin x)}{\sin x} = 1]
  (using Standard Result 1)
• [\lim_{x \to 0} \dfrac{\sin x}{x} = 1]

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Step 3: Multiply the results

[\lim_{x \to 0} \dfrac{\ln(1+\sin x)}{x} = 1 \cdot 1 = 1]

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7. Important Observations

1. Always look for [\ln(1+\text{something})][ → 0]
2. Convert everything to standard forms
3. Use known trigonometric limits when needed
4. Never substitute directly when form is [\dfrac{0}{0}]
5. Write the standard result before using it

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8. Key Standard Results (To Remember)

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Practice Questions with Step-by-Step Solutions

Question 1. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+4x)}{x}]

Step-by-Step Solution:

Step 1: Check the form
As [x → 0],

Numerator → [\ln(1) = 0]

Denominator → [0]
So the form is [\dfrac{0}{0}] (indeterminate)

Step 2: Recall the standard result
[\lim_{x \to 0} \dfrac{\ln(1+ax)}{x} = a]

Step 3: Compare with the given expression
Here, [a = 4]

Final Answer:
[\boxed{4}]

Question 2. Find [\lim_{x \to 0} \dfrac{\ln(1+7x)}{2x}]

Step-by-Step Solution:

Step 1: Identify the indeterminate form
As [x → 0], the form is [\dfrac{0}{0}]

Step 2: Separate the constant in denominator

[\dfrac{\ln(1+7x)}{2x} = \dfrac{1}{2} \cdot \dfrac{\ln(1+7x)}{x}]

Step 3: Apply the standard result

[\lim_{x \to 0} \dfrac{\ln(1+7x)}{x} = 7]

Step 4: Multiply constants

[\dfrac{1}{2} \times 7 = \dfrac{7}{2}]

Final Answer:
[\boxed{\dfrac{7}{2}}]

Question 3. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+\sin x)}{x}]

Step-by-Step Solution:

Step 1: Observe the form
As [x → 0],

[\sin x → 0]

So numerator → [\ln(1) = 0]
Form = [\dfrac{0}{0}]

Step 2: Multiply and divide by [\sin x]

[\dfrac{\ln(1+\sin x)}{x}
= \dfrac{\ln(1+\sin x)}{\sin x} \cdot \dfrac{\sin x}{x}]

Step 3: Evaluate each limit separately

[\lim_{x \to 0} \dfrac{\ln(1+\sin x)}{\sin x} = 1]
(using [\lim_{t \to 0} \dfrac{\ln(1+t)}{t} = 1])

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

Step 4: Multiply results

[1 × 1 = 1]

Final Answer:
[\boxed{1}]

Question 4. Find [\lim_{x \to 0} \dfrac{\ln(1+3x)}{\tan x}]

Step-by-Step Solution:

Step 1: Identify the indeterminate form
As [x → 0], form is [\dfrac{0}{0}]

Step 2: Rewrite the expression

[\dfrac{\ln(1+3x)}{\tan x}
= \dfrac{\ln(1+3x)}{x} \cdot \dfrac{x}{\tan x}]

Step 3: Evaluate each limit

[\lim_{x \to 0} \dfrac{\ln(1+3x)}{x} = 3]

[\lim_{x \to 0} \dfrac{x}{\tan x} = 1]

Step 4: Multiply results

[3 × 1 = 3]

Final Answer:
[\boxed{3}]

Question 5. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+x^2)}{x^2}]

Step-by-Step Solution:

Step 1: Identify the form
As [x → 0], numerator → [\ln(1) = 0], denominator → [0]

Step 2: Compare with standard limit

[\lim_{t \to 0} \dfrac{\ln(1+t)}{t} = 1]

Step 3: Substitute [t = x^2]
Since [x → 0 ⇒ x^2 → 0]

Final Answer:
[\boxed{1}]

Question 6. Find [\lim_{x \to 0} \dfrac{\ln(1+5\sin x)}{x}]

Step-by-Step Solution:

Step 1: Form check
As [x → 0], [\sin x → 0] ⇒ form [\dfrac{0}{0}]

Step 2: Multiply and divide by [\sin x]

[\dfrac{\ln(1+5\sin x)}{x}
= \dfrac{\ln(1+5\sin x)}{\sin x} \cdot \dfrac{\sin x}{x}]

Step 3: Evaluate limits

[\lim_{x \to 0} \dfrac{\ln(1+5\sin x)}{\sin x} = 5]

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

Step 4: Multiply

[5 × 1 = 5]

Final Answer:
[\boxed{5}]

Question 7. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+\tan x)}{x}]

Step-by-Step Solution:

Step 1: As [x → 0], [\tan x → 0]
Form = [\dfrac{0}{0}]

Step 2: Multiply and divide by [\tan x]

[\dfrac{\ln(1+\tan x)}{x}
= \dfrac{\ln(1+\tan x)}{\tan x} \cdot \dfrac{\tan x}{x}]

Step 3: Use standard limits

[\lim_{x \to 0} \dfrac{\ln(1+\tan x)}{\tan x} = 1]

[\lim_{x \to 0} \dfrac{\tan x}{x} = 1]

Final Answer:
[\boxed{1}]

Question 8. Find [\lim_{x \to 0} \dfrac{\ln(1+2x)}{3x}]

Step-by-Step Solution:

Step 1: Rewrite

[\dfrac{1}{3} \cdot \dfrac{\ln(1+2x)}{x}]

Step 2: Apply standard result

[\lim_{x \to 0} \dfrac{\ln(1+2x)}{x} = 2]

Step 3: Multiply

[\dfrac{1}{3} × 2 = \dfrac{2}{3}]

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

Question 9. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+x^4)}{x^4}]

Step-by-Step Solution:

Step 1: Recognize standard form

[\lim_{t \to 0} \dfrac{\ln(1+t)}{t} = 1]

Step 2: Here, [t = x^4]

Step 3: Since [x → 0 ⇒ x^4 → 0]

Final Answer:
[\boxed{1}]

Question 10. Find [\lim_{x \to 0} \dfrac{\ln(1+6x)}{x}]

Step-by-Step Solution:

Step 1: Check the form → [\dfrac{0}{0}]

Step 2: Apply standard logarithmic result

[\lim_{x \to 0} \dfrac{\ln(1+ax)}{x} = a]

Step 3: Here, [a = 6]

Final Answer:
[\boxed{6}]