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Real Analysis
· Examples✏️ Worked Examples — Step by Step
E1
EasyCompute lim x→3 (x² − 9) / (x − 3)
💡 Factor numerator.
👁 Show Solution
(x² − 9)/(x − 3) = (x + 3)(x − 3)/(x − 3) = x + 3 → at x=3, limit = 6. ✅ Answer: 6
✅6
E2
MediumIs f(x) = |x| differentiable at x = 0?
💡 Check left and right derivatives.
👁 Show Solution
Left derivative: −1. Right derivative: +1. They differ → not differentiable at x=0. ✅ Answer: No
✅No
E3
ChallengingDoes the sequence aₙ = 1/n converge? If yes, to what?
💡 As n → ∞, what does 1/n approach?
👁 Show Solution
lim n→∞ (1/n) = 0. Yes — converges to 0. ✅ Answer: 0
✅0