Physics and Mathematics
Limits of the form 1 ^ infinity: Example 1
Practice Questions with Step-by-Step Solutions
Exponential Limits of the Form
[\lim_{x \to \infty} (1 + \dfrac{a}{x})^x]
Question 1. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x]
Step-by-Step Solution:
Step 1: Identify the form
As [x → ∞],
[\dfrac{2}{x} → 0]
So the expression becomes of the form:
[(1 + 0)^{\infty}] = [1^{\infty}] (indeterminate)
Step 2: Recall the standard result
[\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x = e^a]
Step 3: Compare with given expression
Here, [a = 2]
Final Answer:
[e^{2}]
Question 2. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{5}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{5}{x} → 0]
Form obtained: [1^{\infty}]
Step 2: Use the standard exponential limit
[\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x = e^a]
Step 3: Identify the value of [a]
Here, [a = 5]
Final Answer:
[e^{5}]
Question 3. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{1}{x} → 0]
Step 2: Recognize this as the basic standard limit
[\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x = e]
Final Answer:
[e]
Question 4. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{7}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{7}{x} → 0]
Step 2: Expression matches the form
[(1 + \dfrac{a}{x})^x]
Step 3: Identify [a = 7]
Final Answer:
[e^{7}]
Question 5. Evaluate [\lim_{x \to \infty} \left(1 – \dfrac{3}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{3}{x} → 0]
Form becomes [1^{\infty}]
Step 2: Rewrite in standard form
[(1 + \dfrac{-3}{x})^x]
Step 3: Apply standard result
[\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x = e^a]
Here, [a = -3]
Final Answer:
[e^{-3}] or [\dfrac{1}{e^{3}}]
Question 6. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{10}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{10}{x} → 0]
Step 2: Identify standard exponential form
Step 3: Here, [a = 10]
Final Answer:
[e^{10}]
Question 7. Evaluate [\lim_{x \to \infty} \left(1 – \dfrac{1}{2x}\right)^x]
Step-by-Step Solution:
Step 1: Rewrite the base
[(1 + \dfrac{-1}{2x})]
Step 2: Match with standard form
We want [(1 + \dfrac{a}{x})^x]
So here:
[a = -\dfrac{1}{2}]
Step 3: Apply standard result
Final Answer:
[e^{-\dfrac{1}{2}}]
Question 8. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{4}{3x}\right)^x]
Step-by-Step Solution:
Step 1: Rewrite coefficient
[(1 + \dfrac{\dfrac{4}{3}}{x})^x]
Step 2: Identify [a = \dfrac{4}{3}]
Step 3: Apply standard limit
Final Answer:
[e^{\dfrac{4}{3}}]
Question 9. Evaluate [\lim_{x \to \infty} \left(1 + \dfrac{\sin 1}{x}\right)^x]
Step-by-Step Solution:
Step 1: Note that [\sin 1] is a constant
Step 2: As [x → ∞],
[\dfrac{\sin 1}{x} → 0]
Step 3: Compare with standard form
[a = \sin 1]
Final Answer:
[e^{\sin 1}]
Question 10. Evaluate [\lim_{x \to \infty} \left(1 – \dfrac{\pi}{x}\right)^x]
Step-by-Step Solution:
Step 1: As [x → ∞],
[\dfrac{\pi}{x} → 0]
Step 2: Write in standard form
[(1 + \dfrac{-\pi}{x})^x]
Step 3: Apply standard result
Final Answer:
[e^{-\pi}]
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