- #1
devanlevin
[(n+5)/(n-1)]^n
i get [infinity/nifinity]^infinity and i don't see anything to change
i get [infinity/nifinity]^infinity and i don't see anything to change
HallsofIvy said:A little bit more accurately, as n goes to infinity (n+5)/(n+1) goes to 1 so this is of the form [itex]1^{\infty}= 1[/itex].
If this weren't in the "precalculus" section, I would recommend using L'Hopital's rule!
Mark44 said:True enough that it's of the indeterminate form [tex]1^\infty[/tex], but it's not necessarily equal to 1. Another limit with this form is lim (1 + 1/n)^n, for n approaching [tex]\infty[/tex]. The limit here is the natural number, e.
Yes! Mark44 essentially gives that! Dividing, (n+5)/(n-1)= 1+ 6/(n-1) so ((n+5)/(n-1))n= (1+ 6/(n-1))n. Since, for n going to infinity, the difference between n and n-1 is negligible, the limit is the same as the limit of (1+ 6/n)n= (1+ (6/n))6(n/6)= (1+ 1/m)6m where m= 6/n. That is [(1+ 1/m)m]6. As Mark44 said, the limit of (1+ 1/m)m is e so the limit of [(1+ 1/m)m]6 is e6devanlevin said:all true, but the limits anwer is e^6
A limit is a fundamental concept in calculus that describes the behavior of a function as its input value approaches a certain point. It is denoted by the symbol "lim" and is used to determine the value that a function approaches as its input value gets closer and closer to a specified point.
A limit exists for a given function if the two-sided limit and the one-sided limits from both directions approach the same value. In other words, the function must have a consistent behavior as the input value gets closer to the specified point.
If the denominator of a fraction approaches infinity, the limit is equal to zero. This is because as the denominator gets larger and larger, the fraction becomes smaller and smaller, approaching zero as its limit.
Yes, algebraic manipulation can be used to find the limit of a fraction. This involves simplifying the fraction by factoring and canceling out common terms in the numerator and denominator. However, it is important to note that this method only works if the limit exists for the given function.
Yes, there are other methods for finding the limit of a fraction, such as using L'Hôpital's rule or the squeeze theorem. These methods are particularly useful for more complex fractions or functions where algebraic manipulation may not be possible.