Continuity and Differentiability
ICSE · Class 12 · Mathematics
Flashcards for Continuity and Differentiability — ICSE Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Explore the full setSolve: Find lim x→0 (sin x / x).
Answer
Step 1: Use the standard limit. Step 2: Substitute directly from the known result. lim x→0 (sin x / x) = 1. Answer: 1…
Solve: Find lim x→0 (tan x / x).
Answer
Step 1: Use the standard limit. lim x→0 (tan x / x) = 1. Answer: 1…
Apply the limit lim x→0 ((e^x - 1)/x). What is the value?
Answer
Step 1: Use the standard result. lim x→0 ((e^x - 1)/x) = 1. Answer: 1…
Solve: Find lim x→0 (log(1+x) / x).
Answer
Step 1: Use the standard limit. lim x→0 (log(1+x) / x) = 1. Answer: 1…
A function has lim x→c- f(x) = 4 and lim x→c+ f(x) = 4, but f(c) = 6. Is it continuous at x = c?
Answer
Step 1: Check the continuity condition: lim x→c f(x) = f(c). Step 2: The two one-sided limits are equal, so the full limit is 4. Step 3: Compare with f(c) = 6. Since 4 ≠ 6, the function is not continu…
Find the value of k so that f(x) = { (x^2 - 4)/(x - 2) if x ≠ 2, k if x = 2 } is continuous at x = 2.
Answer
Step 1: Simplify for x ≠ 2. (x^2 - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2) = x + 2. Step 2: Find the limit at x = 2. lim x→2 (x + 2) = 4. Step 3: For continuity, f(2) = k must equal the limit. So, k = 4.
Solve: Find the point of discontinuity of f(x) = (|x|/x) for x ≠ 0, with f(0) = 0.
Answer
Step 1: For x > 0, |x|/x = 1. Step 2: For x < 0, |x|/x = -1. Step 3: At x = 0, f(0) = 0. Step 4: The left and right limits at 0 are different: lim x→0- f(x) = -1, lim x→0+ f(x) = 1. Step 5: Since the …
When do you use lim x→c f(x) = f(c)? Give a worked example.
Answer
Use this when checking continuity at a point. Example: f(x) = 2x + 3 at x = 1. Step 1: Find f(1) = 2(1) + 3 = 5. Step 2: Find the limit as x→1. lim x→1 (2x + 3) = 5. Step 3: Compare. lim x→1 f(x) = f(…
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