Suppose a , b are real numbers such that lim x → 0 e a x − b x − 1 sin 2 x = 2 1 . Then

The denominator has expansion ( a − b ) x + 2 a 2 x 2 + O ( x 3 ) , while sin 2 x = x 2 + O ( x 4 ) . A limit of 1/2 requires a = b and 2/ a 2 = 1/2 . Hence a = b = ± 2 .

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Mathematics - Limits Question with Solution | TestHub

MathematicsLimitsValue of unknownEasy2 minQB

MathematicsEasymultiple choice

Suppose $a, b$ are real numbers such that $lim_{x \to 0} \frac{sin ^{2} x}{e ^{a x} - b x - 1} = \frac{1}{2}$ . Then

Options:(select one or more)

Answer:

A, B

Solution:

The denominator has expansion $(a - b) x + \frac{a ^{2}}{2} x^{2} + O (x^{3})$ , while $sin^{2} x = x^{2} + O (x^{4})$ . A limit of $1/2$ requires $a = b$ and $2/ a^{2} = 1/2$ . Hence $a = b = \pm 2$ .

Stream:JEESubject:MathematicsTopic:LimitsSubtopic:Value of unknown

⏱ 2mℹ️ Source: QB

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