Find a,b so lim[x->infty](cbrt{ax^3+bx^2+1}-log_5(3^x+4^x+5^2)-4th-rt{x^4+4})=1

[Roger.Robert]

Find a,b \(\displaystyle \in \)R

\(\displaystyle \lim _{x\to \infty }\left(\sqrt[3]{ax^3+bx^2+1}-\log _5\left(3^x+4^x+5^x\right)-\sqrt[4]{x^4+4}\right)=1\)

 

[Steven G]

Find a,b \(\displaystyle \in \)R

\(\displaystyle \lim _{x\to \infty }\left(\sqrt[3]{ax^3+bx^2+1}-\log _5\left(3^x+4^x+5^x\right)-\sqrt[4]{x^4+4}\right)=1\)

The 2nd (and 3rd) term goes to infinity as x goes to infinity so this limit dne. So the answer is that there are no such a and b that makes the limit equal to 1.

 

[Roger.Robert]

The 2nd (and 3rd) term goes to infinity as x goes to infinity so this limit dne. So the answer is that there are no such a and b that makes the limit equal to 1.

If a=8 ==> 2x-2x=infinity-infinity ==> there's a finite solution

 

[Steven G]

If a=8 ==> 2x-2x=infinity-infinity ==> there's a finite solution

Try a=8 and b=12