Strange limit of a utility fonction

asked 2023-05-11 23:21:27 +0200

Cyrille gravatar image

updated 2023-05-12 02:33:57 +0200

dan_fulea gravatar image

According to my knowledge (and wikipedia) the utility fonction defined

var('x a b γ')
U_5 = lambda x, a, b, γ: ((1-γ)/γ)*(((a*x)/(1-γ))+b)^γ

is such that

$$\lim_{\gamma\rightarrow 0}U_5(x, 1, b, \gamma) = \log(x)$$


limU = lim(U_5(x, 1, b, γ), γ=0)
show(LatexExpr(r"\lim_{\gamma\rightarrow 0}U_5(x, 1, b, γ) = "), limU)

returns $\infty$. Strange. Should I commit an error ?

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I have slightly edited the code, then put in in the right format, so that the post can be digested fluently.

Let us use some letters that can be easily typed, and some special values for $x,b$:

sage: var('c');
sage: U_5(2, 1, 6, c)
-(c - 1)*(-2/(c - 1) + 6)^c/c

Let us now take the limit from this expression for $c\to 0$. Which is the limit? Why should it be $\log 2$?

(Wikipedia comes with easy links, maybe inserting the link would make the readers go there, and not guess for themselves which variable should be specialized to one to get a $\log x$-limit.)

dan_fulea gravatar imagedan_fulea ( 2023-05-12 02:39:46 +0200 )edit