Find the asymptotic lower bound of n³ + n²

[bob]

You see the lower bound for that cubic plus square expression starts with the biggest term right away. I always tell you the cubed part dominates everything else when n gets large. You can drop the square term because it fades in comparison. And the whole thing grows no slower than n to the third power. Perhaps you notice how constants and smaller powers get left behind fast.

I want you to picture n increasing step by step. You test small values first then jump to bigger ones. The cubed term pulls the total upward without stopping. But the square adds only a little extra that shrinks relatively. Now you realize the lower bound locks in at that cubic level. Or maybe you try to prove it by dividing the expression by n cubed.

You end up with one plus one over n which approaches one. I know this shows the function stays above some constant times n cubed. And that constant works for all large enough n. Perhaps you wonder why we bother with lower bounds at all. You already grasp upper bounds from other talks we had.

But lower bounds pin down the minimum speed of growth. I see you nodding when the dominant term clicks in your head. You focus only on the highest power for these bounds. And smaller powers never change the asymptotic picture. Now the square term becomes irrelevant for the lower estimate.

You try subtracting the square and still see the cubic remains. I keep stressing that the lower bound equals the cubic order. Perhaps another way is to factor out n cubed from the start. You get n cubed times one plus one over n again. And the factor stays bounded away from zero eventually.

I recall you asked about similar expressions last month. You compared them by peeling off lower powers. But this case stays straightforward because the cubic leads. Now imagine plotting both the function and the bound. You watch the function stay above the line for big n.

And that visual confirms the lower bound holds tight. You might think about edge cases with tiny n but they do not matter here. I always skip those when discussing asymptotics with you. Perhaps you practice by picking random large n values yourself.

You calculate both sides and compare ratios. And the ratio never drops below a positive number. I think this builds your intuition quicker than any proof. You see the square term lose influence as n climbs.

But the cubic keeps driving the growth from below. Now you connect this to other problems like sorting costs. You realize many algorithms hit similar cubic behaviors. And lower bounds help you know when something cannot run faster.

I want you to remember that ignoring lower terms simplifies everything. You apply the same idea across polynomials of any degree. Perhaps you extend it to logs or other functions later.

You already handle basic cases like this one with ease. And the expression n cubed plus n squared fits the pattern perfectly. I see the lower bound sitting exactly at the cubic order.

You confirm it by the limit method we discussed before. But sometimes direct comparison works even better for you. Now the answer settles as n to the third power in order notation.

You practice rewriting the expression to isolate that term. And the rest vanishes in the limit. I keep these talks going because you pick up the ideas fast.

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