Physics and Mathematics
Exponential Limits Result 2: Example 1
Standard Results Used (Repeated Every Time)
-  [\lim_{x→0} \dfrac{a^{x} – 1}{x} = \log_{e} a]
-  [\lim_{x→0} \dfrac{\sin x}{x} = 1]
Practice Questions – Mixed Exponential Forms
Question 1. Evaluate: [\lim_{x→0} \dfrac{2^{\sin x} – 1}{x}]
Step-by-Step Solution:
As [x→0]:
[\sin x → 0]
Hence [2^{\sin x} → 2^{0} = 1]
So the given limit is of the form [\dfrac{0}{0}].
Multiply and divide by [\sin x]:
[\dfrac{2^{\sin x} – 1}{x} ][= \dfrac{2^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply standard limits:
[\lim_{\sin x→0} \dfrac{2^{\sin x} – 1}{\sin x} ][= \log_{e} 2]
[\lim_{x→0} \dfrac{\sin x}{x} = 1]
Multiply the results:
[\log_{e} 2 \cdot 1 = \log_{e} 2]
Final Answer:
[\log_{e} 2]
Question 2. Evaluate: [\lim_{x→0} \dfrac{5^{\sin x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [\sin x→0], so the form is [\dfrac{0}{0}].
Rewrite:
[\dfrac{5^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply standard limits:
[\dfrac{5^{\sin x} – 1}{\sin x} → \log_{e} 5]
[\dfrac{\sin x}{x} → 1]
Multiply:
[\log_{e} 5]
Final Answer:
[\log_{e} 5]
Question 3. Evaluate: [\lim_{x→0} \dfrac{3^{2\sin x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [\sin x→0], hence [2\sin x→0].
The form is [\dfrac{0}{0}].
Rewrite:
[\dfrac{3^{2\sin x} – 1}{2\sin x} \cdot \dfrac{2\sin x}{x}]
Evaluate each part:
[\dfrac{3^{2\sin x} – 1}{2\sin x} → \log_{e} 3]
[\dfrac{2\sin x}{x} → 2]
Multiply:
[2 \log_{e} 3]
Final Answer:
[2 \log_{e} 3]
Question 4. Evaluate: [\lim_{x→0} \dfrac{7^{\sin x} – 1}{2x}]
Step-by-Step Solution:
Rewrite:
[\dfrac{1}{2} \cdot \dfrac{7^{\sin x} – 1}{x}]
Further rewrite:
[\dfrac{1}{2} \cdot \dfrac{7^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply limits:
[\dfrac{7^{\sin x} – 1}{\sin x} → \log_{e} 7]
[\dfrac{\sin x}{x} → 1]
Multiply:
[\dfrac{1}{2} \log_{e} 7]
Final Answer:
[\dfrac{1}{2} \log_{e} 7]
Question 5. Evaluate: [\lim_{x→0} \dfrac{10^{\sin x} – 1}{x}]
Step-by-Step Solution:
The form is [\dfrac{0}{0}].
Rewrite using multiplication:
[\dfrac{10^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply standard results.
Final Answer:
[\log_{e} 10]
Question 6. Evaluate: [\lim_{x→0} \dfrac{4^{3\sin x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [3\sin x→0].
Rewrite:
[\dfrac{4^{3\sin x} – 1}{3\sin x} \cdot \dfrac{3\sin x}{x}]
Apply limits:
[\dfrac{4^{3\sin x} – 1}{3\sin x} → \log_{e} 4]
[\dfrac{3\sin x}{x} → 3]
Multiply:
[3 \log_{e} 4]
Final Answer:
[3 \log_{e} 4]
Question 7. Evaluate: [\lim_{x→0} \dfrac{a^{\sin x} – 1}{x}], where [a > 0]
Step-by-Step Solution:
This is the general mixed form.
Rewrite:
[\dfrac{a^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply standard limits.
Final Answer:
[\log_{e} a]
Question 8. Evaluate: [\lim_{x→0} \dfrac{9^{\sin x} – 1}{5x}]
Step-by-Step Solution:
Rewrite:
[\dfrac{1}{5} \cdot \dfrac{9^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply limits.
Final Answer:
[\dfrac{1}{5} \log_{e} 9]
Question 9. Evaluate: [\lim_{x→0} \dfrac{e^{\sin x} – 1}{x}]
Step-by-Step Solution:
Rewrite:
[\dfrac{e^{\sin x} – 1}{\sin x} \cdot \dfrac{\sin x}{x}]
Apply standard limits:
[\dfrac{e^{y} – 1}{y} → 1]
[\dfrac{\sin x}{x} → 1]
Final Answer:
[1]
Question 10. Evaluate: [\lim_{x→0} \dfrac{2^{\tan x} – 1}{x}]
Step-by-Step Solution:
As [x→0], [\tan x→0].
Rewrite:
[\dfrac{2^{\tan x} – 1}{\tan x} \cdot \dfrac{\tan x}{x}]
Apply limits:
[\dfrac{2^{\tan x} – 1}{\tan x} → \log_{e} 2]
[\dfrac{\tan x}{x} → 1]
Final Answer:
[\log_{e} 2]
Final Student Takeaway
Whenever you see
[\dfrac{a^{\sin x} – 1}{x}]
- Immediately split it into two standard limits
- Convert it into logarithmic × trigonometric form
- Apply known results step by step
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