Class 12 Continuity And Differentiability · Long Answer3 Markspiecewise_continuity
Class 12 Maths · Continuity And Differentiability · Long Answer #124
Let f(x)=(e^(1/x)−1)/(e^(1/x)+1) for x≠0, and f(0)=−1. Show discontinuity at 0.
✅ Step-by-Step Solution
Final Answer: Discontinuous at 0.
- As x→0⁺, e^(1/x)→∞. Divide numerator and denominator by e^(1/x) to obtain right limit 1.
- As x→0⁻, e^(1/x)→0, so the left limit is −1.
- The one-sided limits differ, so the two-sided limit does not exist, even though f(0) matches the left limit.
- Therefore f is discontinuous at 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] As x→0⁺, e^(1/x)→∞. Divide numerator and denominator by e^(1/x) to obtain right limit 1.
- [1 mark] As x→0⁻, e^(1/x)→0, so the left limit is −1.
- [1 mark] The one-sided limits differ, so the two-sided limit does not exist, even though f(0) matches the left limit.
- [0.5 marks] Therefore f is discontinuous at 0.
Master Chapter 5: Continuity And Differentiability
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
← Previous Question
Let f(x)=2x−1 for x<1/2 and f(x)=3−6x for x≥1/2. Discuss…
Next Question →For f(x)=x²+3x+a when x≤1 and bx+2 when x>1, find a,b for…
More Practice Questions in Continuity And Differentiability
Related Questions in Continuity And Differentiability
Class 12 Continuity And Differentiability · Mcq Single Correct
For k<0, define f(x)=(1−cos(kx))/(x sin x) for nonzero x near 0, and f(0)=1/2. Find k for…
Class 12 Continuity And Differentiability · Mcq Single Correct
Near π/2, f(x)=k cos x/(π−2x) for x≠π/2 and f(π/2)=3. Find k for continuity.
Class 12 Continuity And Differentiability · Mcq Single Correct
Find the full set of real inputs excluded from the natural domain of cot x (its poles).
Class 12 Continuity And Differentiability · Mcq Single Correct
How many real inputs are excluded from the original formula f(x)=(x−1)/[x(x²−1)]?
Class 12 Continuity And Differentiability · Mcq Single Correct
For x≠0, f(x)=(e^(3x)−e^(−5x))/x, and f(0)=k. Find k for continuity at 0.
Class 12 Continuity And Differentiability · Mcq Single Correct
Let f(x)=x/|x| for x<0 and f(x)=−1 for x≥0. What is its complete continuity set?