HELP PLEASE - harder EXTENSION 1 (limits) (1 Viewer)

[cssftw]

Have no idea how to do this:

limit as n-->(infinity) OF (3^n + 5^n)^(1/n) --> just graphed it - it's a weird graph, the answer is 5 - can someone please show working out

help appreciated

 

[Carrotsticks]

Consider the expression:



We know that if we omit the 3^n, we acquire the inequality:



We know that 3 < 5, so we have 3^n < 5^n.

If we were to put this into our expression, we acquire:



So combining our two inequalities yields:



So taking the limit as n approaches infinity, we acquire:




And so by the Squeeze Law, we have:



This is actually a specific case of this:

 

[RealiseNothing]

Consider the expression:



We know that if we omit the 3^n, we acquire the inequality:



We know that 3 < 5, so we have 3^n < 5^n.

If we were to put this into our expression, we acquire:



So combining our two inequalities yields:



So taking the limit as n approaches infinity, we acquire:




And so by the Squeeze Law, we have:



This is actually a specific case of this:

Love Squeeze Theorem/Law.

 

[Shadowdude]

Love Squeeze Theorem/Law.

*cough*Pinching Theorem*cough*

But yes, this kinda proves once again that Harder 3u can be the most difficult topic in MX2.

 

[barbernator]

just put (3^99 + 5^99)^1/99 in calc so much easier hahaha... nah that law is awesome

 

[deswa1]

just put (3^99 + 5^99)^1/99 in calc so much easier hahaha... nah that law is awesome

This is actually a good way to test whether you are correcct. Ideally though, use a bigger number

 

[barbernator]

This is actually a good way to test whether you are correcct. Ideally though, use a bigger number

my calc gives 5 by using that anyway