text: ## Evaluating $\lim_{x\to 0} x^x$
  
  $$x ^{x } = e^{\ln x^x }$$ $$\lim_{x\rightarrow 0} x^x = \;?$$ I need to find the limit of x to the power of x as x approaches to 0 using l’Hopital’s rule. From previous part there is a hint that I should use the first equation somehow, however I am … Read more
  
  ## Does y=(−1)xy = (-1)^x where x∈ℝx∈ℝ, change exponentially?
  
  Is y = (-1)^x an exponential curve, or just a sinusoidal one, can it be said to change exponentially as with positive exponents? I’m sure W/A showed this as being sinusoidal with an integer period. But now shows the answer as -1. From the answers I understand that their is no definitive definition but their … Read more
  
  ## Exponent xx tending to infinity
  
  When an expression has x tending to infinity and it’s an exponent, like: lim is there some way to get a solid value for this limit (hopefully 0, but at this stage I’m not picky)? Answer Your limit is 0. I would advise that you read about this in your book. It is impossible that … Read more
  
  ## Find the Limit: limx→2+e3/2(2−x)\lim_{x\to2^{+}}e^{3/2(2-x)}
  
  \lim_{x\to2^{+}}e^{3/2(2-x)} Properties of the Natural Exponential Function: The exponential function f(x)=e^x is an increasing continuous function with domain \mathbb R and range (0, \infty). Thus e^x>0 for all x. Also \lim_{x\to-\infty}e^x=0 and the \lim_{x\to\infty} e^x=\infty So the x-axis is a horizontal asymptote of f(x)= e^x. How do I utilize this definition to solve this problem? … Read more
  
  ## Two exponential terms equation solution
  
  Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using Lambert W function. But can I get a general solution of this equation? $$A_1 \exp(B_1x) + A_2 \exp(B_2x) + A_3x + 1 = 0.$$ I searched a lot but did not get an … Read more
  
  ## Compute limx→0ex−1x\lim\limits_{x \to 0} \frac{e^x-1}{x} without using derivatives [duplicate]
  
  This question already has answers here: Proving that limx→0ex−1x=1 (7 answers) Closed 6 years ago. How to compute limx→0ex−1x without using derivatives? Every method I can think of gives me some indeterminate form. Answer Set  ex−1=y, so  x=ln(1+y) and  y→0 for  x→0 Now you have:  limx→0ex−1x=limy→0yln(1+y)=limy→01ln(1+y)y=limy→01ln(1+y)1y Putting  1y=t, for  y→0,t→∞ , so you have:  limt→∞1ln(1+1t)t=1ln(e)=1 … Read more
  
  ## The value of the series \frac{e^{-n}}{n}\frac{e^{-n}}{n}
  
  I have already found that the series \sum_{n=1}^{+\infty} \frac{e^{-n}}{n} converges. However, I want to find its value, but I don’t know how to do that. I’ve tried several things but without success. Need your help ! 🙂 Answer HINT: Let f(x)=\sum_{n\ge 1}\frac{x^n}n; then f'(x)=\sum_{n\ge 1}x^{n-1}=\sum_{n\ge 0}x^n\;. You know a closed form for f'(x), and you … Read more
  
  ## Solving $ax=e^{bx}$ [duplicate]
  
  This question already has answers here: How can one find the zeroes of $f(x)=ae^{bx}+cx+d$? (3 answers) Closed 3 years ago. I’ve been asked by a friend to help him solve this equation, but since we couldn’t find the right answer, I thought about posting it here. Firstly, I thought about derivating both sides and get: … Read more
  
  ## Solve tet=t+2t\, e^t=t+2 for t>0t>0.
  
  I need to solve the following equation: tet=t+2, for t>0. Do I need to use Lambert W function, or there is some other method? Thanks! PS. I know that any equation of the form A+Bt+Clog(D+Et)=0 has solutions which, if they exist, can be expressed using Lambert function. But I do not think that tet=t+2 can … Read more
  
  ## How to derive ∞∑k=01k!\sum\limits_{k=0}^\infty \frac{1}{k!} from lim\lim\limits_{n\to\infty}\left(1+\frac{1}{n}\right)^n
  
  Is it possible to derive \begin{align*} e=\sum\limits_{k=0}^\infty \frac{1}{k!} \end{align*} from \begin{align*} e=\lim\limits_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n \end{align*} ? Thank you:)! Answer If you use the binomial expansion you get \left(1+\frac 1n\right)^n=1+n\cdot \frac 1n+\frac {n(n-1)}{2!}\left(\frac 1n\right)^2+\frac {n(n-1)(n-2)}{3!}\left(\frac 1n\right)^3\dots==1+1+\frac 1{2!}\left(1-\frac 1n\right)+\frac 1{3!}\left(1-\frac 1n\right)\left(1-\frac 2n\right)+\dots I will leave you to conclude. AttributionSource : Link , Question Author : Kevin Meier , … Read more
explanation: The provided text contains multiple math questions and answers related to limits, exponential functions, and series. Each section presents a specific problem and, in some cases, a solution or guidance on how to solve it.
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distilabel_metadata.content: ## Evaluating $\lim_{x\to 0} x^x$
  
  $$x ^{x } = e^{\ln x^x }$$ $$\lim_{x\rightarrow 0} x^x = \;?$$ I need to find the limit of x to the power of x as x approaches to 0 using l’Hopital’s rule. From previous part there is a hint that I should use the first equation somehow, however I am … Read more
  
  ## Does y=(−1)xy = (-1)^x where x∈ℝx∈ℝ, change exponentially?
  
  Is y = (-1)^x an exponential curve, or just a sinusoidal one, can it be said to change exponentially as with positive exponents? I’m sure W/A showed this as being sinusoidal with an integer period. But now shows the answer as -1. From the answers I understand that their is no definitive definition but their … Read more
  
  ## Exponent xx tending to infinity
  
  When an expression has x tending to infinity and it’s an exponent, like: lim is there some way to get a solid value for this limit (hopefully 0, but at this stage I’m not picky)? Answer Your limit is 0. I would advise that you read about this in your book. It is impossible that … Read more
  
  ## Find the Limit: limx→2+e3/2(2−x)\lim_{x\to2^{+}}e^{3/2(2-x)}
  
  \lim_{x\to2^{+}}e^{3/2(2-x)} Properties of the Natural Exponential Function: The exponential function f(x)=e^x is an increasing continuous function with domain \mathbb R and range (0, \infty). Thus e^x>0 for all x. Also \lim_{x\to-\infty}e^x=0 and the \lim_{x\to\infty} e^x=\infty So the x-axis is a horizontal asymptote of f(x)= e^x. How do I utilize this definition to solve this problem? … Read more
  
  ## Two exponential terms equation solution
  
  Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using Lambert W function. But can I get a general solution of this equation? $$A_1 \exp(B_1x) + A_2 \exp(B_2x) + A_3x + 1 = 0.$$ I searched a lot but did not get an … Read more
  
  ## Compute limx→0ex−1x\lim\limits_{x \to 0} \frac{e^x-1}{x} without using derivatives [duplicate]
  
  This question already has answers here: Proving that limx→0ex−1x=1 (7 answers) Closed 6 years ago. How to compute limx→0ex−1x without using derivatives? Every method I can think of gives me some indeterminate form. Answer Set  ex−1=y, so  x=ln(1+y) and  y→0 for  x→0 Now you have:  limx→0ex−1x=limy→0yln(1+y)=limy→01ln(1+y)y=limy→01ln(1+y)1y Putting  1y=t, for  y→0,t→∞ , so you have:  limt→∞1ln(1+1t)t=1ln(e)=1 … Read more
  
  ## The value of the series \frac{e^{-n}}{n}\frac{e^{-n}}{n}
  
  I have already found that the series \sum_{n=1}^{+\infty} \frac{e^{-n}}{n} converges. However, I want to find its value, but I don’t know how to do that. I’ve tried several things but without success. Need your help ! 🙂 Answer HINT: Let f(x)=\sum_{n\ge 1}\frac{x^n}n; then f'(x)=\sum_{n\ge 1}x^{n-1}=\sum_{n\ge 0}x^n\;. You know a closed form for f'(x), and you … Read more
  
  ## Solving $ax=e^{bx}$ [duplicate]
  
  This question already has answers here: How can one find the zeroes of $f(x)=ae^{bx}+cx+d$? (3 answers) Closed 3 years ago. I’ve been asked by a friend to help him solve this equation, but since we couldn’t find the right answer, I thought about posting it here. Firstly, I thought about derivating both sides and get: … Read more
  
  ## Solve tet=t+2t\, e^t=t+2 for t>0t>0.
  
  I need to solve the following equation: tet=t+2, for t>0. Do I need to use Lambert W function, or there is some other method? Thanks! PS. I know that any equation of the form A+Bt+Clog(D+Et)=0 has solutions which, if they exist, can be expressed using Lambert function. But I do not think that tet=t+2 can … Read more
  
  ## How to derive ∞∑k=01k!\sum\limits_{k=0}^\infty \frac{1}{k!} from lim\lim\limits_{n\to\infty}\left(1+\frac{1}{n}\right)^n
  
  Is it possible to derive \begin{align*} e=\sum\limits_{k=0}^\infty \frac{1}{k!} \end{align*} from \begin{align*} e=\lim\limits_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n \end{align*} ? Thank you:)! Answer If you use the binomial expansion you get \left(1+\frac 1n\right)^n=1+n\cdot \frac 1n+\frac {n(n-1)}{2!}\left(\frac 1n\right)^2+\frac {n(n-1)(n-2)}{3!}\left(\frac 1n\right)^3\dots==1+1+\frac 1{2!}\left(1-\frac 1n\right)+\frac 1{3!}\left(1-\frac 1n\right)\left(1-\frac 2n\right)+\dots I will leave you to conclude. AttributionSource : Link , Question Author : Kevin Meier , … Read more
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text contains multiple math questions and answers related to limits, exponential functions, and series. Each section presents a specific problem and, in some cases, a solution or guidance on how to solve it.
  
  Label: multiple-question-and-answer