For a ≥ 0 , let L = lim x → ∞ ( x 2 + 4 a x + 2 ) ( a x 3 + 1 ) / ( x 2 + 1 ) . Which statements are true?

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Mathematics - Continuity - Differentiability Question with Solution | TestHub

MathematicsContinuity - DifferentiabilityContinuity- MiscellaneousHard2 minQB

MathematicsHardmultiple choice

For $a \geq 0$ , let $L = lim_{x \to \infty} (\frac{a x + 2}{x ^{2} + 4})^{(a x^{3} + 1) / (x^{2} + 1)}$ . Which statements are true?

Options:(select one or more)

Answer:

A, B, C

Solution:

For $a = 0$ , logarithmic evaluation gives $L = 1$ . For $a = 1$ , the base is $1 + 2/ x + O (x^{- 2})$ and the exponent is $x + o (x)$ , so $L = e^{2}$ . For fixed $0 < a < 1$ , the base tends to $a < 1$ while the exponent tends to infinity, so $L = 0$ .

Stream:JEESubject:MathematicsTopic:Continuity - DifferentiabilitySubtopic:Continuity- Miscellaneous

⏱ 2mℹ️ Source: QB

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