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Câu 22 trang 151 SGK Đại số và Giải tích 11 Nâng cao

Cho hàm số
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Lựa chọn câu để xem lời giải nhanh hơn
 Cho hàm số \(f\left( x \right) = \cos {1 \over x}\) và hai dãy số \(\left( {x{'_n}} \right),\left( {x{"_n}} \right)\) với \(x_n' = {1 \over {2n\pi }},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x''_n= {1 \over {\left( {2n + 1} \right){\pi \over 2}}}\)

LG a

Tìm giới hạn của các dãy số  \(\left( {x_n'} \right),\left( {x_n"} \right),\left( {f\left( {x_n'} \right)} \right)\) và \(\left( {f\left( {x_n"} \right)} \right)\)

Lời giải chi tiết:

Ta có:

\(\eqalign{
& \lim x_n' = \lim {1 \over {2n\pi }} = 0 \cr 
& \lim x''_n = \lim {1 \over {\left( {2n + 1} \right){\pi \over 2}}} = 0 \cr 
& \lim f\left( {x{'_n}} \right) = \lim \cos 2n\pi = 1 \cr 
♏& \lim f\left( {x{"_n}} \right) = \lim \cos \left( {2n + 1} \right){\pi \over 2} = 0 \cr} \)

LG b

Tồn tại hay không  \(\mathop {\lim }\limits_{x \to 0} \cos {1 \over x}?\)

Lời giải chi tiết:

Do hai dãy \((x'_n)\) và \((x''_n)\) đều tiến đến \(0\) nhưng \(\lim f\left( {x{'_n}} \right) \ne \lim f\left( {x''{_n}} \right)\) nên theo định nghĩa giới hạn hàm số tại một điểm, không tồn tại \(\mathop {\lim }\limits_{x \to 0} \cos {1 \over x}\).

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