SOLUTION: limit x->0 (e^x^2 - e^x^3)/3x^2 please show steps. thanks.

Algebra.Com
Question 1154021: limit x->0 (e^x^2 - e^x^3)/3x^2
please show steps. thanks.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
Answer by ikleyn(52847)   (Show Source): You can put this solution on YOUR website!
.

Start from standard expansion of Calculus for 

     = 1 + t +  +  + . . .               (1)



Substitute one time  t = x^2  into expansion  (1)  to get

    e^(x^2) = 1 + x^2 +  +  + . . .         (2)



Substitute next time  t = x^3  into expansion  (1)  to get

    e^(x^3) = 1 + x^3 +  +  + . . .         (3)



Subtract (3) from (2). You will get

    e^(x^2) - e^^(x^3) = x^2 - x^3 + the terms of the converging series with degrees of  "x"  higher than or equal to 4.


Therefore

     = 1 - x + the terms of the converging series with degrees of  "x"  higher than or equal to 2.     (4)


Thus, when you take the limit at x--> 0 of  ,  you will get  .     ANSWER


The rest of the terms contribute 0 (zero) to this limit.

Solved.

------------------

I wrote this post, because arguments of @MathLover1 are   I N V A L I D.


Had I answered this question as @MathLover1 did it,  with expressions      on the way,  I would loose my reputation  FOREVER
in the eyes of my teachers.



RELATED QUESTIONS

e^x2 - 2 = x^3 - x Please show the steps.... (answered by solver91311)
Y=(e^(x)-e^(-x))/2 please show me the steps to solve,... (answered by jsmallt9)
show that the integration from ( 0 to infinity ) for ( cos 3x / ( x^2+1)^2 )= 2pi/e^3 (answered by Fombitz)
Please solve e^x = 2 using... (answered by josmiceli,nycsharkman,jim_thompson5910)
log(base4)2^((e^x)(sinx)) Could you show me the steps worked out... (answered by josmiceli)
What is the left-hand limit of f(x)= |x-3|/(x-3) as x approaches 3? a. 1 b. 2 c. 0... (answered by ikleyn)
e^x+2=3 (answered by jsmallt9)
limit of e^x/x as x approaches... (answered by MathLover1)
integral calculus 1. integrate the dx/e^x-e^-x 2. integrate the sec^2xdx/8+sec^2x 3. (answered by Alan3354)