Physics and Mathematics
L Hospitals Rule: Example 1
Practice Questions with Step-by-Step Solutions
Question 1. Evaluate [\lim_{x \to 1} \dfrac{x^2 – 1}{x – 1}]
Step-by-Step Solution:
Step 1: Check the form
As [x → 1]:
Numerator → [1 − 1 = 0]
Denominator → [1 − 1 = 0]
Form = [\dfrac{0}{0}]
Step 2: Apply L’Hospital’s Rule
Numerator derivative: [\dfrac{d}{dx}(x^2 – 1) = 2x]
Denominator derivative: [\dfrac{d}{dx}(x – 1) = 1]
Step 3: Evaluate the limit
[\lim_{x \to 1} 2x = 2]
Final Answer:
[\boxed{2}]
Question 2. Find [\lim_{x \to 0} \dfrac{x – \tan x}{x^3}]
Step 1: Check the form
As [x → 0]:
Numerator → [0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: First application of L’Hospital’s Rule
Numerator derivative: [1 − \sec^2 x]
Denominator derivative: [3x^2]
New limit:
[\lim_{x \to 0} \dfrac{1 – \sec^2 x}{3x^2}]
Step 3: Simplify numerator
[1 − \sec^2 x = -\tan^2 x]
So,
[\lim_{x \to 0} \dfrac{-\tan^2 x}{3x^2}]
Step 4: Rewrite
[\lim_{x \to 0} -\dfrac{1}{3} \left(\dfrac{\tan x}{x}\right)^2]
Step 5: Evaluate
[\left(\dfrac{\tan x}{x}\right) → 1]
Final Answer:
[\boxed{-\dfrac{1}{3}}]
Question 3. Evaluate [\lim_{x \to 0} \dfrac{e^x – 1 – x}{x^2}]
Step 1: Check the form
As [x → 0]:
Numerator → [0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: First application of L’Hospital’s Rule
Numerator derivative: [e^x − 1]
Denominator derivative: [2x]
Step 3: Still [\dfrac{0}{0}] → Apply again
Numerator derivative: [e^x]
Denominator derivative: [2]
Step 4: Evaluate
[\lim_{x \to 0} \dfrac{e^x}{2} = \dfrac{1}{2}]
Final Answer:
[\boxed{\dfrac{1}{2}}]
Question 4. Find [\lim_{x \to 0} \dfrac{\sin 3x}{e^x – 1}]
Step 1: Check the form
As [x → 0]:
[\sin 3x → 0]
[e^x − 1 → 0]
Form = [\dfrac{0}{0}]
Step 2: Apply L’Hospital’s Rule
Numerator derivative: [3\cos 3x]
Denominator derivative: [e^x]
Step 3: Evaluate
[\lim_{x \to 0} \dfrac{3\cos 3x}{e^x} = \dfrac{3}{1}]
Final Answer:
[\boxed{3}]
Question 5. Evaluate [\lim_{x \to \infty} \dfrac{x}{e^x}]
Step 1: Check the form
As [x → ∞]:
Numerator → [∞]
Denominator → [∞]
Form = [\dfrac{\infty}{\infty}]
Step 2: Apply L’Hospital’s Rule
Numerator derivative: [1]
Denominator derivative: [e^x]
Step 3: Evaluate
[\lim_{x \to \infty} \dfrac{1}{e^x} = 0]
Final Answer:
[\boxed{0}]
Question 6. Find [\lim_{x \to 0} \dfrac{\tan x – \sin x}{x^3}]
Step 1: Check the form
As [x → 0]:
Numerator → [0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: First L’Hospital application
Numerator derivative: [\sec^2 x − \cos x]
Denominator derivative: [3x^2]
Step 3: Still [\dfrac{0}{0}] → Apply again
Numerator derivative: [2\sec^2 x \tan x + \sin x]
Denominator derivative: [6x]
Step 4: Still [\dfrac{0}{0}] → Apply again
Numerator derivative: [2(2\sec^2 x \tan^2 x + \sec^4 x) + \cos x]
Denominator derivative: [6]
Step 5: Evaluate
At [x = 0]:
[\tan 0 = 0], [\sec 0 = 1], [\cos 0 = 1]
So numerator → [2(0 + 1) + 1 = 3]
Final Answer:
[\boxed{\dfrac{1}{2}}]
Question 7. Evaluate [\lim_{x \to 0} \dfrac{\ln(1+x) – x}{x^2}]
Step 1: Check the form
As [x → 0]:
Numerator → [0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: First application
Numerator derivative: [\dfrac{1}{1+x} − 1]
Denominator derivative: [2x]
Step 3: Simplify numerator
[\dfrac{1 – (1+x)}{1+x} = \dfrac{-x}{1+x}]
Step 4: New limit
[\lim_{x \to 0} \dfrac{-x}{(1+x)(2x)}]
Step 5: Cancel [x]
[\lim_{x \to 0} \dfrac{-1}{2(1+x)}]
Step 6: Evaluate
[\dfrac{-1}{2}]
Final Answer:
[\boxed{-\dfrac{1}{2}}]
Question 8. Find [\lim_{x \to 0} \dfrac{e^{2x} – e^x}{x}]
Step 1: Check the form
As [x → 0]:
Numerator → [1 − 1 = 0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: Apply L’Hospital’s Rule
Numerator derivative: [2e^{2x} − e^x]
Denominator derivative: [1]
Step 3: Evaluate
At [x = 0]:
[2 − 1 = 1]
Final Answer:
[\boxed{1}]
Question 9. Evaluate [\lim_{x \to 0} \dfrac{\sin x – x\cos x}{x^3}]
Step 1: Check the form
As [x → 0]:
Numerator → [0]
Denominator → [0]
Form = [\dfrac{0}{0}]
Step 2: First application
Numerator derivative: [\cos x − (\cos x − x\sin x)]
Denominator derivative: [3x^2]
Simplify numerator:
[\cos x − \cos x + x\sin x = x\sin x]
Step 3: New limit
[\lim_{x \to 0} \dfrac{x\sin x}{3x^2}]
Step 4: Simplify
[\lim_{x \to 0} \dfrac{\sin x}{3x}]
Step 5: Apply L’Hospital again
Numerator derivative: [\cos x]
Denominator derivative: [3]
Step 6: Evaluate
[\dfrac{1}{3}]
Final Answer:
[\boxed{\dfrac{1}{3}}]
Question 10. Find [\lim_{x \to \infty} \dfrac{x^2}{e^x}]
Step 1: Check the form
As [x → ∞]:
Numerator → [∞]
Denominator → [∞]
Form = [\dfrac{\infty}{\infty}]
Step 2: First application
Numerator derivative: [2x]
Denominator derivative: [e^x]
Step 3: Still [\dfrac{\infty}{\infty}] → Apply again
Numerator derivative: [2]
Denominator derivative: [e^x]
Step 4: Evaluate
[\lim_{x \to \infty} \dfrac{2}{e^x} = 0]
Final Answer:
[\boxed{0}]
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