What is the Limit at Infinity for (2^x-5^x) / (3^x+5^x)?

[terryds]

Homework Statement

lim x->∞ (2^x-5^x) / (3^x+5^x)

Choices :

a. -1
b. -2/3
c. 1
d. 6
e. 25

2. The attempt at a solution

Hmmm.. I really have no idea about this.. This is an unusual problem..
Please tell me...

 

[Raghav Gupta]

Homework Statement

lim x->∞ (2^x-5^x) / (3^x+5^x)

Choices :

a. -1
b. -2/3
c. 1
d. 6
e. 25

2. The attempt at a solution

Hmmm.. I really have no idea about this.. This is an unusual problem..
Please tell me...

Try dividing numerator and denominator by something.

 

[Delta2]

Hint: nominator can be written as ##5^x((\frac{2}{5})^x-1)##. Also denominator can be written in a very similar way. Also i think you know that for any ##0<a<1## it is ##\lim\limits_{x \to +\infty}a^x=0##. If you use all this info i believe you should be able to find the correct answer.

 

[terryds]

Try dividing numerator and denominator by something.

Hmm..
I have no idea..
By what something ?
If I divide numerator and denominator by x, it will just make things more complicated
Since the 2^x/x can't be simplified more... (The bad thing is the x is the exponent, not in the number)

 

[Raghav Gupta]

Hmm..
I have no idea..
By what something ?
If I divide numerator and denominator by x, it will just make things more complicated
Since the 2^x/x can't be simplified more... (The bad thing is the x is the exponent, not in the number)

Use Delta² hints and try writing the denominator.

 

[terryds]

Hint: nominator can be written as ##5^x((\frac{2}{5})^x-1)##. Also denominator can be written in a very similar way. Also i think you know that for any ##0<a<1## it is ##\lim\limits_{x \to +\infty}a^x=0##. If you use all this info i believe you should be able to find the correct answer.

lim x->∞ 5^x((2/5)^x-1) / (5^x ((3/5)^x + 1))
lim x->∞ (2/5)^x - 1 / ((3/5)^x + 1)
-1/1 = -1

Okay, I've got that the answer is a. -1
Yeah, thanks for your hint

 

[HallsofIvy]

Notice that this was the same as "divide both numerator and denominator by [itex]5^x[/itex]".