How to evaluate this limit?

Question

#lim xrarrprop (1+1/(2x^2))^(x^2)#

322 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-24T05:24:45

# e^(1/2), or, sqrte#.

Recall that, #lim_(t to oo) (1+1/t)^t=e#.

Accordingly, the reqd. lim.#=lim_(x to oo)(1+1/(2x^2))^(x^2)#,

#=lim_(2x^2 to oo){(1+1/(2x^2))^(2x^2)}^(1/2)#.

Now, because of the continuity of the function #x^(1/2)#,

#"The Reqd. lim."=[lim_(2x^2 to oo){(1+1/(2x^2))^(2x^2)}]^(1/2)#,

#=e^(1/2), or, sqrte#.