JEEAdvanced
Let L = lim_{x->0} ( arcsin(x) / x )^(1/x^2). The value is L = e^a…
Question
Let L = lim_{x->0} ( arcsin(x) / x )^(1/x^2). The value is L = e^a for a rational a = m/n in lowest terms with m, n positive integers and gcd(m,n)=1. Find m + n.
✓ Verified answer: 7checked by our engine — not a guess
Step-by-step solution
Use arcsin(x) = x + x^3/6 + O(x^5), so arcsin(x)/x = 1 + x^2/6 + O(x^4).
Then ln( arcsin(x)/x ) = x^2/6 + O(x^4), and (1/x^2)·ln( arcsin(x)/x ) = 1/6 + O(x^2) -> 1/6.
Hence L = e^(1/6), so a = 1/6, m = 1, n = 6, and m + n = 7.
Final answer7
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Limits solutions
Let L = lim_{x->0} ( (1+x)^(1/x) - e + (e x)/2 ) / x^2. The value of…JEE · MathLet L = lim_{n->infinity} n·( e - (1 + 1/n)^n ). The value can be…JEE · MathLet L = lim_{x->0} ( cosh(x)·cos(x) - 1 ) / x^4. Writing L = m/n in…JEE · MathLet L = lim_{x->0} ( (1+x)^(1/x) / e )^(1/x). The value is L = e^a…JEE · MathLet L = lim_{x->0} ( (1 + x^2)^(1/3) - cos(x) ) / ( x·sin(x) ).…JEE · MathLet L = lim_{x->0} ( (1 + x + x^2)^(1/x) - e·(1 + x/2) ) / ( e·x^2 ).…JEE · MathCompute L = lim_{x->infinity} ( (x^3 + 3 x^2)^(1/3) - (x^2 - 2…JEE · MathLet L = lim_{x->0} ( (1 - cos x)(1 - cos 2x)(1 - cos 3x) ) / x^6.…JEE · Math