Exams › JEE Main › Maths

If lim_(x→ ∞) (1+(a)/(x)+(b)/(x²))^(2x)=e², then which of the following describes a and b?

1. a=1 and b=2
2. a=1, with b any real number
3. a any real number, and b=2
4. a any real number, and b any real number

Correct answer: a=1, with b any real number

Solution

The limit expression approaches the form of the exponential function, and for it to equal e², the coefficient of the linear term in the expansion must be 1, which means a must be 1. The term involving b does not affect the leading behavior as x approaches infinity, allowing b to be any real number.

Related JEE Main Maths questions

• Evaluate the limit as x approaches 0: ((1+tan x)/(1+sin x))^(csc x).
• Evaluate the limit as x approaches 0 of sin([x−3]) divided by [x−3], where [.] denotes the greatest integer function.
• Evaluate the limit as x approaches 0: (sin(x⁴) − x⁴ cos(x⁴) + x²⁰) / (x⁴(e^(2x⁴) − 1 − 2x⁴)).
• Evaluate the limit lim_(n→∞)(n³-8)/(n³+8). What is its value?
• If lim_(x→∞) (1+(λ)/(x)+(μ)/(x²))^(2x)=e², which of the following values of λ and μ is correct?
• Consider the following statements: Statement 1: If lim_(x→α)(sin(f(x)))/(x-α), where f(x)=ax²+bx+c, exists as a finite nonzero number, then lim_(x→α)(1/f(x)-1)/(1/f(x)+1) does not exist. Statement 2: The limit lim_(x→α)(sin(f(x)))/(x-α) can be finite only in the indeterminate form 0/0.

⚔️ Practice JEE Main Maths free + battle 1v1 →