LIMIT THESE SUMS OR THEY'LL GET UNLIMITED

1. lim_{nâ†’âˆž} (4ⁿ+5ⁿ)^1/n=

2. lim_{nâ†’âˆž} ( xⁿ + yⁿ )/(xⁿ - yⁿ) where x>y>1 is equal to

3)lim_{xâ†’ - âˆž} {âˆš(x² - x + 1) - ax - b} = 0. then (a,b) is equal to

4)lim _xâ†’0 (cos(sin x) - cos x)/x⁴ is equal to

PLEASE SOLVE THESE WITH PROPER EXPLANATION

#Mathematics #Calculus

2 Answers

for first one take 5ⁿ cmmon , then (4/5)ⁿ â†’0 as nâ†’âˆž so limit is 5

second is similar , take xⁿ common , y/x bcomes fraction, and (y/x)ⁿ tends to zero as n â†’âˆž

for third , rationalise first, then put y = -x , and then check wat shud be done to keep the limit finite i.e 0 here.

for fourth use the half angle formula of 1- cosx = 2sin²(x/2)

i.e 1-cos(sinx) = 2sin²((sinx)/2)

ans. 1 )is 5 and 2)is 1.

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