Giải SBT Toán 11 (Kết nối tri thức) Bài 16: Giới hạn của hàm số

Cho hàm số $f\left(x\right)=\left(\begin{array}{c}x neu x>1\\ 2 neu x=1\\ 1 neu x<1\end{array}\right)$ . Hàm số f(x) có giới hạn khi x → 1 không?

Tính các giới hạn sau:

a) $\underset{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">\to </span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">lim</span></span>}}\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">4</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">1</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">3</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}$ ;

b) $\underset{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">\to </span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">lim</span></span>}}\frac{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">3</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">3</span></span>}}{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">3</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">1</span></span>}}$ ;

c) $\underset{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">\to </span></span>{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}^{<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">lim</span></span>}}\frac{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">5</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">6</span></span>}}{{\left(\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}\right)}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}}$ ;

d) $\underset{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">\to </span></span>{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">0</span></span>}}^{<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">lim</span></span>}}\frac{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:12px;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}<span\; style="font-size:18px;"><span\; style="font-size:12px;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">2</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:12px;">x</span></span>}}$ .

Tìm a để hàm số $f\left(x\right)=\left(\begin{array}{c}{x}^2+ax neu x>3\\ 3x^2+1 neu x\le 3\end{array}\right)$ có giới hạn khi x → 3.

Tìm các số thực a và b sao cho $\underset{x\to 1}{\lim }\frac{2x^2-ax+1}{x^2-3x+2}=b$ .

Cho hàm số $f\left(x\right)=\frac{\sqrt{x^2-x+2}}{x}$ . Tính:

a) $\underset{x\to +\infty }{\lim }f\left(x\right)$ ;

b) $\underset{x\to -\infty }{\lim }f\left(x\right)$ .

Tính giới hạn $\underset{x\to +\infty }{\lim }\left(1-x\right)\left(1-x^2\right)\left(1-x^3\right)$ .

Cho hàm số $g\left(x\right)=\sqrt{x^2+2x}-\sqrt{x^2-1}-2m$ với m là tham số. Biết $\underset{x\to +\infty }{\lim }g\left(x\right)=0$ , tìm giá trị của m.

Cho m là một số thực. Biết $\underset{\mathrm{<span\; style="font-size:18px;">x</span>}<span\; style="font-size:18px;">\to </span><span\; style="font-size:18px;">-</span><span\; style="font-size:18px;">\infty </span>}{\mathrm{<span\; style="font-size:18px;">lim</span>}}\left(\left(\mathrm{<span\; style="font-size:18px;">m</span>}<span\; style="font-size:18px;">-</span>\mathrm{<span\; style="font-size:18px;">x</span>}\right)\left(\mathrm{<span\; style="font-size:18px;">m</span>}\mathrm{<span\; style="font-size:18px;">x</span>}<span\; style="font-size:18px;">+</span>\mathrm{<span\; style="font-size:18px;">1</span>}\right)\right)<span\; style="font-size:18px;">=</span><span\; style="font-size:18px;">-</span><span\; style="font-size:18px;">\infty </span>$ . Xác định dấu của m.

Cho hàm số $f\left(x\right)=\frac{{\sin }^{2}x}{x^2}$ . Chứng minh rằng $\underset{x\to +\infty }{\lim }f\left(x\right)=0$ .

Một đơn vị sản xuất hàng thủ công ước tính chi phí để sản xuất x đơn vị sản phẩm là C(x) = 2x + 55 (triệu đồng).

a) Tìm hàm số f(x) biểu thị chi phí trung bình để sản xuất mỗi đơn vị sản phẩm.

b) Tính $\underset{x\to +\infty }{\lim }f\left(x\right)$ . Giới hạn này có ý nghĩa gì?