Question

Find the limit......

Answer 1

39. Lt(x→ ∞) ( 1 + 2/3x)^(7x+2)/5

first of all we check which type of limit appear here,
put x = ∞
we see Limit is in the form of 1^∞ ,

now,

e^Lt (x→∞) (7x + 2)/5 log{1 + 2/3x }

we know,
Lt (f(x) → a) log(1 + f(x))/f(x)
where f(a) = 0 then,
Lt(f(x)→a ) log(1 + f(x))/f(x) = 1

use this concept here ,

e^Lt(x→∞)(7x +2)/5 {log(1 + 2/3x)/(2/3x) }×(2/3x)

e^Lt(x→∞) (7x +2)/5 × 2/3x

dividing by x in both numerator or denominator .

= e^Lt(x→∞) {7 + 2/x )× 2/5×3

= e^14/15