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Continuity and Differentiability — Flashcards

ICSE · Class 12 · Mathematics

29 flashcards for Continuity and Differentiability (ICSE Class 12 Mathematics) to test yourself on key terms and facts.

105 questions29 flashcards9 formulas & key relations5 concepts

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A graph of a continuous function at a specific point, showing the limit approaching the function value, illustrating that the function is unbroken and defined at that point.
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29 Flashcards·
Standard LimitsContinuity at a PointRemovable DiscontinuityDiscontinuity TypesContinuity Formula Application
Card 1Standard Limits

Solve: 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…

Card 2Standard Limits

Solve: Find lim x→0 (tan x / x).

Answer

Step 1: Use the standard limit. lim x→0 (tan x / x) = 1. Answer: 1…

Card 3Standard Limits

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…

Card 4Standard Limits

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…

Card 5Continuity at a Point

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…

Card 6Removable Discontinuity

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.

Card 7Discontinuity Types

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 …

Card 8Continuity Formula Application

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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Frequently Asked Questions

What are the important topics in Continuity and Differentiability for ICSE Class 12 Mathematics?
Key topics in Continuity and Differentiability include Limits and One-Sided Limits, Continuity at a Point and on an Interval, Algebra of Continuous Functions, Differentiability at a Point and One-Sided Derivatives. Study these first, then practise questions on each for the ICSE Class 12 board exam.
How many flashcards are available for Continuity and Differentiability?
There are 29 flashcards for Continuity and Differentiability covering key definitions, facts and ideas. A few sample cards are shown on this page.
How should I revise Continuity and Differentiability for the ICSE Class 12 board exam?
Learn the core ideas first, then work through the 105 practice questions on Continuity and Differentiability. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

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