If limx→11+ax+bx2ax-1 =e-3, find the values of ordered pair (a,b) - Maths - Limits and Derivatives

Question

I f   lim x → 1 1 + a x + b x 2 a x - 1   = e - 3 ,
  f i n d   t h e   v a l u e s   o f   o
r d e r e d   p a i r   ( a , b )

Aarushi Mishra · Accepted Answer

a-1≤a≤a1 x-1≤1 x≤1 x2 x-1≤2 x≤2
x n x-1≤n x≤n xAdding1 x-1+2 x-1+3 x-1+ +n x-1≤1
x+2 x+3 x+ +n x≤1 x+2 x+3 x+ +n x1 x+2 x+3 x+ +n x-n≤1
x+2 x+3 x+ +n x≤1 x+2 x+3 x+ +n x1 x+2 x+3 x+ +n
x-nn2≤1 x+2 x+3 x+ +n xn2≤1 x+2 x+3 x+ +n xn21 x+2 x+3
x+ +n xn2-1n≤1 x+2 x+3 x+ +n xn2≤1 x+2 x+3 x+ +n
xn21n1n+2n+3n+ +nnx-1n≤1 x+2 x+3 x+ +n xn2≤1n1n+2n+3n+
+nnx1n∑r=1nrnx-1n≤1 x+2 x+3 x+ +n
xn2≤1n∑r=1nrnxlimn→∞1n∑r=1nrnx-limn→∞1n≤limn→∞1
x+2 x+3 x+ +n
xn2≤limn→∞1n∑r=1nrnxlimn→∞1n∑r=1nrnx+0≤limn→∞1
x+2 x+3 x+ +n
xn2≤limn→∞1n∑r=1nrnx⇒limn→∞1
x+2 x+3 x+ +n
xn2=limn→∞1n∑r=1nrnxUsing limn→∞ 1n∑r=ab frn=∫limn→∞anlimn→∞bn fxdxlimn→∞1
x+2 x+3 x+ +n
xn2=∫limn→∞1nlimn→∞nn xdx=∫01 x dx=x2201=12

I f lim x → 1 1 + a x + b x 2 a x - 1 = e - 3 , f i n d t h e v a l u e s o f o r d e r e d p a i r ( a , b ) .

$I f \lim_{x \to 1} {(1 + a x + b x^{2})}^{\frac{a}{x - 1}} = e^{- 3}, f i n d t h e v a l u e s o f o r d e r e d p a i r (a, b) .$