limits and tangents

[Jaina]

I've got the same type of problem about seven times in my homework assignment for the night and I just can't figure it out:

lim h-->0 (h^3 + 3h^2 + 3h - 3) / h

How do I get that stinking "h" out of the denominator? I can do it with square root signs, or the ones where I have to factor an "h" out of the numerator, but with stuff like this I'm stuck. I know there's an easier way to do this problem with derivatives, but I don't get credit for the work if I don't do it the long way. :roll: So what do I do?

 

[stapel]

Since there is no common factor of "h" in the numerator, there is no way to "cancel off" the "h" in the denominator.

Was this the original limit provided to you, or does this represent how far you got? If the latter, then please provide the full text of the exercise, and all the steps you have tried. Thank you.

Eliz.

 

[Jaina]

Sorry. Here's the original problem:
Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
h(t)= t^3 + 3t, (1, 4)

And here's what I've done:
lim h-->0 (((t+h)^3 + 3(t+h)) - (t^3 + 3t)) / h
lim h-->0 (((1+h)^3 + 3(1+h)) - (1 + 3)) / h
lim h-->0 (((1+h)^3 + 3(1+h)) - 4 / h
lim h-->0 ((1+2h+h^2)(1+h) + 3 + 3h - 4) / h
lim h-->0 (h^3 + 3h^2 + 3h -3) / h

Maybe I've been playing the wrong games with the numerator?

 

[stapel]

I'm not sure what you're doing there in the middle...? I'll rename the function as "f(t)", to avoid the confusion of "h" standing for two different things.

. . . . .[f(t + h) - f(t)] / [h]

. . . . .[f(1 + h) - f(1) / [h]

. . . . .[( (1 + h)³ + 3(1 + h) ) - (1³ + 3(1))] / [h]

. . . . .[1 + 3h + 3h² + h³ + 3 + 3h - 1 - 3] / [h]

. . . . .[6h + 3h² + h³] / [h]

. . . . .[(h)(6 + 3h + h²] / [h]

Now cancel the h's, and evaluate the result at h = 0.

Eliz.

Edit: Re-inserted a dropped term.

 

[Jaina]

Well. I'm not entirely sure what I was doing in the middle, either. That certainly makes a lot more sense than... whatever it was that I was trying to do.

One last thing: when you go from this:
[1 + 3h + 3h2 + h3 + 3 + 3h - 1 - 3] / [h]

to this:
[3h + 3h2 + h3] / [h]

shouldn't it be 6h instead of just 3h? Both of the 3h's in the first section are positive--how does one disappear?

 

[stapel]

...shouldn't it be 6h instead of just 3h?

Yes, you're quite right. It can be distressingly easy to drop terms when they're surrounded by lots of coding symbols, and I did just that in this case.

Thank you for catching the error. I have updated the previous post to be (closer to) correct. :wink:

Eliz.