- #1
temaire
- 279
- 0
Homework Statement
[tex]lim_{x\rightarrow\infty}(\frac{3}{5})^{x}[/tex]
2. The attempt at a solution
I know the answer is 0 by using the calculator, but how do I solve this algebraically?
praharmitra said:how do u get a zero? the limit is towards infinity. Try dividing num and denominator with x. Then apply the limits...
praharmitra said:for the first question... the algebraic way...? for any number a < 1 limit (x->inf) a^x = 0 always. and 3/5 < 1.
No, not at all. Another way to say this is to factor x from the numerator and denominator. Since lim x/x = 1, as x --> infinity, what you're left with has the same limit.temaire said:If I divide the numerator and denominator by x, wouldn't I end where I started?
The algebraic solution to the limits problem lim (3/5)^x as x approaches infinity is 0. This can be found by applying the exponential limit rule, which states that for a limit of the form lim a^x as x approaches infinity, the limit is equal to 0 if a is between 0 and 1.
The limit of (3/5)^x as x approaches infinity can be determined by using the exponential limit rule. This rule states that if the base of the exponential expression is between 0 and 1, the limit will be equal to 0. In this case, the base (3/5) is between 0 and 1, so the limit is equal to 0.
Yes, this limits problem can also be solved using the graphical approach. By graphing the function (3/5)^x, it can be seen that as x approaches infinity, the graph approaches the x-axis, indicating a limit of 0.
The exponent approaching infinity in this limits problem signifies that the input values are becoming increasingly large. This is important because it allows us to see the behavior of the function at extremely large input values and determine its limit.
Yes, the algebraic solution to this limits problem can be applied to other exponential expressions as long as the base is between 0 and 1. This is a general rule for finding the limit of exponential functions as the exponent approaches infinity.