Questions: un → +∞, vn → -∞ e wn → 2 lim (un × vn) lim (wn/vn) lim (vn)^3 lim sqrt(1/un)

Transcript text: \[ \boldsymbol{u}_{\boldsymbol{n}} \rightarrow+\infty, v_{n} \rightarrow-\infty \text { e } w_{\boldsymbol{n}} \rightarrow 2 \] $\lim \left(u_{n} \times v_{n}\right)$ $\lim \left(\frac{w_{n}}{v_{n}}\right)$ $\lim \left(v_{n}\right)^{3}$ $\lim \sqrt{\frac{1}{u_{n}}}$

Solution

Solution Steps

To solve these limit problems, we need to consider the behavior of each sequence as $ n $ approaches infinity.

1.5. The limit of the product $ u_n \times v_n $ as $ n \to \infty $ involves $ u_n \to +\infty $ and $ v_n \to -\infty $. The product of a sequence going to positive infinity and a sequence going to negative infinity will go to negative infinity.

1.6. The limit of the fraction $ \frac{w_n}{v_n} $ as $ n \to \infty $ involves $ w_n \to 2 $ and $ v_n \to -\infty $. Dividing a constant by a sequence going to negative infinity will result in the limit going to zero.

1.7. The limit of $ (v_n)^3 $ as $ n \to \infty $ involves $ v_n \to -\infty $. Cubing a sequence going to negative infinity will result in the limit going to negative infinity.

Step 1: Analyze the Limit of $ u_n \times v_n $

Given that $ u_n \to +\infty $ and $ v_n \to -\infty $, the product $ u_n \times v_n $ will tend towards negative infinity. This is because multiplying a sequence that diverges to positive infinity by a sequence that diverges to negative infinity results in a sequence that diverges to negative infinity.

Step 2: Analyze the Limit of $ \frac{w_n}{v_n} $

Given that $ w_n \to 2 $ and $ v_n \to -\infty $, the fraction $ \frac{w_n}{v_n} $ will tend towards zero. This is because dividing a constant by a sequence that diverges to negative infinity results in a sequence that converges to zero.

Step 3: Analyze the Limit of $ (v_n)^3 $

Given that $ v_n \to -\infty $, the cube $ (v_n)^3 $ will tend towards negative infinity. This is because cubing a sequence that diverges to negative infinity results in a sequence that also diverges to negative infinity.