Question

Let \[ f(x)=\lim_{\theta\to 0}\frac{\cos(\pi x)-x^{2/\theta}\sin(x-1)}{1-x^{2/\theta}(x-1)}. \] Statement 1: \(f(x)\) is discontinuous at \(x=1\).
Statement 2: \(f(x)\) is continuous at \(x=-1\).

• A. Both Statements are correct
• B. Both Statements are false
• C. Statement 1 is false and Statement 2 is correct
• D. Statement 1 is correct and Statement 2 is false

Hint:

When limits involve terms like \(x^{1/\theta}\) as \(\theta\to0\), always split the analysis into cases: \(|x|<1,\ |x|>1,\ |x|=1\).

Answer 1

The Correct Option is B

We are given the function f(x) defined by a limit involving a parameter θ. Our task is to examine the continuity of f(x) at x = 1 and x = −1.

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Given function:

f(x) = lim (θ → 0) [ cos(πx) − x^(2/θ) · sin(x − 1) ] / [ 1 − x^(2/θ)(x − 1) ]

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Step 1: Behaviour of x^(2/θ) as θ → 0

• If |x| < 1, then x^(2/θ) → 0.
• If |x| = 1, then x^(2/θ) = 1.
• If |x| > 1, then x^(2/θ) → ∞.

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Step 2: Continuity at x = 1

At x = 1:

x^(2/θ) = 1

Substitute x = 1 into the function:

f(1) = [ cos(π) − sin(0) ] / [ 1 − 0 ]

f(1) = −1

The limit exists and equals the function value.

Therefore, f(x) is continuous at x = 1.

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Step 3: Continuity at x = −1

At x = −1:

(−1)^(2/θ) = 1

Substitute x = −1 into the function:

f(−1) = [ cos(−π) − sin(−2) ] / [ 1 − (−1 − 1) ]

f(−1) = [ −1 + sin(2) ] / 3

The limit exists and equals the function value.

Therefore, f(x) is continuous at x = −1.

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Conclusion:

• f(x) is continuous at x = 1.
• f(x) is continuous at x = −1.

Hence, both statements claiming discontinuity are incorrect.