plane geometry - Does the intersection of two moving curves sweep out a continuous line?

How to display following restrict of features? Answer

Hello expensive customer to our community We will proffer you an answer to this query How to display following restrict of features? ,and the respond will breathe typical by way of documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

How to display following restrict of features?

Permit $ zeta in C_ {c} ^ { infty} (B (0,1)) $. Then as $ M rightarrow infty $
$$ M ^ {n-1} N int _ { Omega} zeta (M x) e ^ {- p N x_ {n}} dx rightarrow frac {1} {p} int _ { mathbb { R} ^ {n-1}} zeta left (x ^ { prime}, 0 privilege) dx ^ { prime} $$
Where $ (x_1, cdots, x_ {n-1}, x_n) = (x ‘, x_n) $, $ 1 and $ N = N (M) $ in order that $ M / N to $ 0 how $ M to infty $.

My try: With Change of variable we’ve $ (x ‘, x_n) = left ( frac {x’} {M}, frac {x_n} {M ^ 2} privilege) $. Jacobi = $ frac {1} {M ^ {n + 1}} $
commence {align *} M ^ {n-1} N int _ { Omega} zeta (M x) e ^ {- p N x_ {n}} d x & = M ^ {n-1} N int_ { Omega} zeta (x ‘, frac {x_n} {M}) e ^ {- pN frac {x_n} {M ^ 2}} frac {1} {M ^ {n + 1}} dx & = frac {N} {M ^ 2} int _ { mathbb {R} ^ {n-1}} int _ {- 1} ^ {1} zeta (x ‘, frac {x_n} {M}) e ^ {- pN frac {x_n} {M ^ 2}} dx_n dx ‘ aim {align *}
I attempted utilizing the combination by components however I could not get the outcome I wanted.

Any ameliorate or counsel is appreciated. Thank you very mighty.

we are going to proffer you the answer to How to display following restrict of features? query through our community which brings all of the solutions from a number of dependable sources.