Exams › JEE Main › Maths

Evaluate the limit as x approaches 0: (sin(x⁴) − x⁴ cos(x⁴) + x²⁰) / (x⁴(e^(2x⁴) − 1 − 2x⁴)).

1. 0
2. −1/6
3. 1/6
4. The limit does not exist

Correct answer: 1/6

Solution

Let u = x^4. Numerator sin u - u cos u + ... ~ u^3/3 = x^12/3. Denominator x^4(e^(2u)-1-2u) ~ x^4 * 2u^2 = 2 x^12. The limit = (1/3)/2 = 1/6.

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 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.
• Evaluate the limit lim_(x→∞) ((3x-4)/(3x+2))^((x+1)/(3)):

⚔️ Practice JEE Main Maths free + battle 1v1 →