Solving limits that approach infinity

Question

Solve limit: $$\lim_{k\to\infty}\frac{4k+3}{2}$$ My approach. $$\lim_{k\to\infty}\frac{(4k+3)}{2} \frac{1/k}{1/k}$$ $$\lim_{k\to\infty}\frac{4+3/k}{2/k}$$ Then, we know that the limit of k as k approaches inf for $$\frac{3}{k}$$ goes to 0 and the same goes for $$\frac{2}{k}$$ however for both limits, they can also be viewed as infinitesmall such that the final answer is $$\frac{4}{0+} = \infty$$

Can somebody verify if I did this correctly? I'm confused as to whether the fraction limits approach 0 or infinitesmall because if it's 0 then my limit is undefined

Answer 1

Why so complicated? $$\lim_{k\to\infty}\frac{4k+3}{2}=\lim_{k\to\infty}(2k+1.5)=2\infty+1.5=\infty$$

Answer 2

Simply note that

$$\frac{4k+3}{2}=2k+\frac32\ge 2k \ge k \to \infty$$

then conclude by squeeze theorem.

Answer 3

Can't you find $k$ such that $$\frac{4k+3}2>1000000\ ?$$

And larger numbers ?

If yes, the expression is unbounded.