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goody1
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Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.
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goody said:Hello everyone, can anybody solve this limit without using L'Hospital's rule? This is really tough one for me, I know I'd use that x = e^lnx.
goody said:I understand that you used Maclaurin series expansion and then the first step behind first equal sign but may I ask how did we get another steps?
goody said:Oh of course, I got it. And in the last step you just divided 3/2 by n because n is infinite number of terms behind dots or why is it like that? Why did you not divide 1 by n as well?
goody said:Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.
Klaas van Aarsen said:It's like this:
$$\Big(1-\frac 12 x^2 + \ldots\Big)\Big(1+\frac 12 (2x)^2 - \ldots\Big)\\
=1\cdot 1 -\frac 12 x^2 \cdot 1+1\cdot \frac 12 (2x)^2 - \frac 12 x^2 \cdot \frac 12 (2x)^2 + \text{ other terms with }x^4\text{ and higher order}\\
=1 + \Big(-\frac 12 + \frac 12(2^2)\Big)x^2 + \ldots \\
= 1+\frac 32 x^2 + \ldots
$$
goody said:Still, now I'm wondering how we got this .
Is it correct? Because I think it should be like that or did I miss something?
Prove It said:So much unneccessary analysis when L'Hospital's Rule is so concise...
Klaas van Aarsen said:The original OP asked explicitly to do it without L'Hospital's Rule.
Prove It said:Oh really?
Klaas van Aarsen said:Ah well, I was kind of happy to see that the OP showed interest in power series expansions.
They are kind of... well... powerful.
The limit of an exponential function is the value that the function approaches as the independent variable (usually denoted as x) gets closer and closer to a certain value. It is also known as the "end behavior" of the function.
To find the limit of an exponential function, you can use the rules of limits, such as the limit laws and L'Hopital's rule. You can also use a graphing calculator or a table of values to estimate the limit.
The limit of an exponential function can be affected by the value of the base (a in the function f(x) = a^x), the value of the exponent (x), and the value that the independent variable is approaching (usually denoted as c).
Yes, the limit of an exponential function can be infinity. This can happen when the base of the exponential function is greater than 1, and the exponent is approaching a very large positive number. In this case, the function will grow without bound as x approaches infinity.
Yes, there are many real-life applications of limits of exponential functions. For example, in finance, the concept of compound interest can be modeled using an exponential function with a limit. In physics, the rate of radioactive decay can also be modeled using an exponential function with a limit.