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Câu 21 trang 151 SGK Đại số và Giải tích 11 Nâng cao

Áp dụng định nghĩa giới hạn
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Áp dụng định nghĩa giới hạn của hàm số, tìm các giới hạn sau :

LG a

 \(\mathop {\lim }\limits_{x \to - 1} {{{x^2} - 3x - 4} \over {x + 1}}\)

Giải chi tiết:

 Với \(x ≠ -1\) ta có  \(f\left( x \right) = {{{x^2} - 3x - 4} \over {x + 1}} = {{\left( {x + 1} \right)\left( {x - 4} \right)} \over {x + 1}} = x - 4\)

Với mọi dãy số (xn🤪) trong khoảng \(\mathbb R\backslash \left\{ { - 1} \right\}\) (tức \(x_n≠ -1, ∀n\)) mà \(\lim\, x_n = -1\) ta có :

\(\lim f\left( x_n \right) = \lim \left( {{x_n} - 4} \right) = - 1 - 4 = - 5\) Vậy  \(\mathop {\lim }\limits_{x \to - 1} {{{x^2} - 3x - 4} \over {x + 1}} = - 5\)

LG b

\(\mathop {\lim }\limits_{x \to 1} {1 \over {\sqrt {5 - x} }}\)

Giải chi tiết:

Tập xác định của hàm số \(f\left( x \right) = {1 \over {\sqrt {5 - x} }}\) là \(D = (-∞ ; 5)\)

Với mọi dãy (xn😼) trong khoảng \(\left( { - \infty {\rm{ }};{\rm{ }}5} \right)\backslash \left\{ 1 \right\}\) sao cho  \(\lim\, x_n = 1\), ta có :

\(\lim f\left( {{x_n}} \right) = \lim {1 \over {\sqrt {5 - {x_n}} }} = {1 \over 2}\) Vậy  \(\mathop {\lim }\limits_{x \to 1}  {1 \over {\sqrt {5 - x} }} = {1 \over 2}\)

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