Show that if lim_(x->a) f(x)=L, then lim_(x->a) cos(f(x))=cos(L).

gaby131o 2022-09-26 Answered
Show that if lim x a f ( x ) = L, then lim x a c o s ( f ( x ) ) = c o s ( L ).
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Answers (2)

Mackenzie Lutz
Answered 2022-09-27 Author has 13 answers
cos ( A ) cos ( B ) = 2 sin ( A + B 2 ) sin ( A B 2 )
Apply this with A = f ( x ) and B = LL. Then
| cos ( f ( x ) ) cos ( L ) | = 2 | sin ( f ( x ) + L 2 ) sin ( f ( x ) L 2 ) | .
Now, recall that | sin ( t ) | | t | , so,
| cos ( f ( x ) ) cos ( L ) | 1 2 | f ( x ) + L | | f ( x ) L | .
And from here, it is easy to conclude a δ ε type proof.
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Ivan Buckley
Answered 2022-09-28 Author has 4 answers
For the sake of completeness,
cos ( a + b ) = cos ( a ) cos ( b ) sin ( a ) sin ( b )
cos ( a b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b )
by substracting
cos ( a + b ) cos ( a b ) = 2 sin ( a ) sin ( b ) .
Now, calling A = a + b and B = a b, gives a = A + B 2 and b = A B 2 .
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