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Limit of $\frac {x^c-c^x} {x^x-c^c}$ as $x \rightarrow c$ My question is: Show that $\lim_ {x \rightarrow c} \frac {x^c-c^x} {x^x-c^c}$ exists and find its value Because the limit is 0 0 I've tried using L'Hopital's rule, but every time I differentiate it I still get 0 0?
Vector derivation of $x^Tx$ - Mathematics Stack Exchange From this it should be clear that $$\frac {d} {dx} x^t x = 2x^t$$ (The transpose is there because the derivative is a map $\mathbb {R}^n\rightarrow\mathbb {R}$, so expressed as a matrix it must have dimension $1\times n$, or alternatively, as a linear map it must live in the dual space to $\mathbb {R}^n$, i e the space of linear maps $\mathbb
Limit of $x^x$ as $x$ tends to $0$ - Mathematics Stack Exchange You have basically two options Allow complex arguments, then the limit doesn't exist, or restrict to positive [or non-negative] arguments Then $\lim\limits_ {x\searrow 0} e^ {x\log x}$ settles the matter
Problem when integrating $e^x x$. - Mathematics Stack Exchange This part looks right: $$\int {\frac {e^x} {x}} \, dx = \frac {e^x} {x} + \frac {e^x} {x^2} + \frac {2e^x} {x^3} + \frac {6 e^x} {x^4} + \frac {24 e^x} {x^5} + \cdots+ \frac {n!e^x} {x^ {n+1}}+ (n+1)!\int \frac {e^x} {x^ {n+1}}$$ When you say "repeating to infinity" you want to take the limit of that in order for your equality to hold, you need $$\lim_n (n+1)!\int \frac {e^x} {x^ {n+1}}=0
Maclaurin expansion of $\arcsin x$ - Mathematics Stack Exchange 26 I'm trying to find the first five terms of the Maclaurin expansion of $\arcsin x$, possibly using the fact that $$\arcsin x = \int_0^x \frac {dt} { (1-t^2)^ {1 2}} $$ I can only see that I can interchange differentiation and integration but not sure how to go about this Thanks!
How to solve $e^x=x$? - Mathematics Stack Exchange I mean, how to solve $e^x=x$ in the complex plane? Is there a solution? I know there is no real solution to this because apparently, $y=x$ and y=$e^x$ have no