Limits of the Form
[\lim_{x→0} \dfrac{a^{x} – 1}{x}]
1. Concept Overview (Very Important for Exams)
When a function involves an exponential expression with base [a] (a > 0, a ≠ 1) and the variable [x] is in the power, direct substitution [x = 0] gives:
- Numerator → [a^{0} − 1 = 0]
- Denominator → [0]
So the limit becomes [\dfrac{0}{0}], which is an indeterminate form.
Hence, standard results are required.
2. Standard Result (Must Be Memorised)
[\boxed{\lim_{x→0} \dfrac{a^{x} – 1}{x} = \log_{e} a}]
Here:
- [\log_{e} a] means natural logarithm of [a]
- Also written as [\ln a]
3. Why This Result Is True
- As [x→0], the curve [y = a^{x}] behaves almost like a straight line near [x = 0]
- The slope of this curve at [x = 0] is exactly [\log_{e} a]
- Hence, the limit measures the rate of change of [a^{x}] at zero
This idea also connects limits with derivatives, which students will study later.
4. Important Variations of the Same Result
From the basic formula, we immediately get:
- [\lim_{x→0} \dfrac{a^{kx} – 1}{x} ][= k \log_{e} a]
- [\lim_{x→0} \dfrac{a^{x} – 1}{bx} ][= \dfrac{\log_{e} a}{b}]
- [\lim_{x→0} \dfrac{a^{f(x)} – 1}{f(x)} ][= \log_{e} a],
provided [f(x) → 0] as [x→0]
Practice Questions with Step-by-Step Solutions
Question 1. Evaluate: [\lim_{x→0} \dfrac{2^{x} – 1}{x}]
Step-by-Step Solution:
As [x→0]:
[2^{x} → 1]
Numerator → [0], Denominator → [0]
So the form is [\dfrac{0}{0}].
Use the standard result:
[\lim_{x→0} \dfrac{a^{x} – 1}{x} = \log_{e} a]
Here, [a = 2].
Final Answer:
[\log_{e} 2]
Question 2. Evaluate: [\lim_{x→0} \dfrac{5^{x} – 1}{x}]
Step-by-Step Solution:
The limit is of the form [\dfrac{0}{0}].
Apply the standard formula directly.
Here, [a = 5].
Final Answer:
[\log_{e} 5]
Question 3. Evaluate: [\lim_{x→0} \dfrac{3^{2x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [2x→0], hence the form is [\dfrac{0}{0}].
Use the modified standard result:
[\lim_{x→0} \dfrac{a^{kx} – 1}{x} = k \log_{e} a]
Here:
[a = 3]
[k = 2]
Substitute values.
Final Answer:
[2 \log_{e} 3]
Question 4. Evaluate: [\lim_{x→0} \dfrac{7^{x} – 1}{2x}]
Step-by-Step Solution:
The expression is of the form [\dfrac{0}{0}].
Rewrite:
[\dfrac{1}{2} \cdot \dfrac{7^{x} – 1}{x}]
Apply the standard limit:
[\dfrac{7^{x} – 1}{x} → \log_{e} 7]
Multiply by [\dfrac{1}{2}].
Final Answer:
[\dfrac{1}{2} \log_{e} 7]
Question 5. Evaluate: [\lim_{x→0} \dfrac{10^{x} – 1}{x}]
Step-by-Step Solution:
The form is [\dfrac{0}{0}].
Apply the standard exponential limit.
Here, [a = 10].
Final Answer:
[\log_{e} 10]
Question 6. Evaluate: [\lim_{x→0} \dfrac{4^{3x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [3x→0], so the form is indeterminate.
Use:
[\lim_{x→0} \dfrac{a^{kx} – 1}{x} = k \log_{e} a]
Here:
[a = 4]
[k = 3]
Final Answer:
[3 \log_{e} 4]
Question 7. Evaluate: [\lim_{x→0} \dfrac{9^{x} – 1}{5x}]
Step-by-Step Solution:
Rewrite:
[\dfrac{1}{5} \cdot \dfrac{9^{x} – 1}{x}]
Apply the standard limit.
Final Answer:
[\dfrac{1}{5} \log_{e} 9]
Question 8. Evaluate: [\lim_{x→0} \dfrac{a^{x} – 1}{x}], where [a > 0]
Step-by-Step Solution:
This is the general standard form.
Directly apply the known result.
Final Answer:
[\log_{e} a]
Key Exam Takeaway
- If the variable is in the power, think logarithm
- This limit always results in [\log_{e} a]
- This result is used repeatedly in:
- Exponential limits
- Logarithmic limits
- Derivatives (later chapters)
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