Finding the Limit of a Fraction with Exponential Terms

[PCSL]

$$\lim_{n\rightarrow ∞}\frac{3^n+2*5^n}{2^n+3*5^n}$$

I tried using l'hospital's rule and got

$$\lim_{n\rightarrow ∞}\frac{3^n*ln(3)+2(5^n*ln5)}{2^n*ln(2))+3(5^n*ln5)}$$

I'm not quite sure if that is the right way to approach this. This problem is an early one in the assignment so I assume that it is simple and I am just missing something obvious.

Thank you!

 

[Mark44]

$$\lim_{n\rightarrow ∞}\frac{3^n+2*5^n}{2^n+3*5^n}$$

I tried using l'hospital's rule and got

$$\lim_{n\rightarrow ∞}\frac{3^n*ln(3)+2(5^n*ln5)}{2^n*ln(2))+3(5^n*ln5)}$$

I'm not quite sure if that is the right way to approach this. This problem is an early one in the assignment so I assume that it is simple and I am just missing something obvious.

Thank you!

A much simpler approach is to factor 5ⁿ out of all terms in the numerator and denominator. Evaluating the limit is pretty easy after that.

 

[PCSL]

A much simpler approach is to factor 5ⁿ out of all terms in the numerator and denominator. Evaluating the limit is pretty easy after that.

so then I'd have

$$\lim_{n\rightarrow ∞} \frac{\frac{3}{5}^n+2}{\frac{2}{5}^n+3}$$

Nevermind, I got it thanks!

 

[Mark44]

Yes, that's right. The key idea is to find the dominant term in the numerator or denominator, which turns out to be 5ⁿ in this problem.

 

[PCSL]

For

$$\lim_{n\rightarrow ∞}(\frac{n+1}{n})^n$$

is l'hoptials rules the correct way to approach it?

 

[Mark44]

No, since L'Hopital's Rule applies to quotients, and that's not what you have here. (The quotient is raised to a nonconstant power.

The usual approach to this type of problem is:
1) Let y = the expression in the limit. Don't include the limit operation.
2) Take the natural log of both sides to get ln y = ln(expression).
3) Use the properties of logs to simplify the right side
4) Take the limit of both sides.
5) Switch the limit and ln operations to get ln(lim y). I.e., this is the log of what you want.

 

[PCSL]

No, since L'Hopital's Rule applies to quotients, and that's not what you have here. (The quotient is raised to a nonconstant power.

The usual approach to this type of problem is:
1) Let y = the expression in the limit. Don't include the limit operation.
2) Take the natural log of both sides to get ln y = ln(expression).
3) Use the properties of logs to simplify the right side
4) Take the limit of both sides.
5) Switch the limit and ln operations to get ln(lim y). I.e., this is the log of what you want.

Really appreciate it.