# Math Help - cal 1 problems limits

1. ## cal 1 problems limits

I can't figure out this one, its driving me crazy. My teacher made this problems up and he call them Interesting problems and sometimes even the tutors struggle with his problems.

Lim 5 + 2x^2 + x^3 / over 3 - x^2 + 2x^3
x->infinity

It has the sign of Infinity but i can't draw it and -> is an arrow as x approaches...

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Evaluate the limit and illustrate each step that involves the use of a limit property.

Lim t^3 - 3t^2 - t + 3 / over t^2 - t - 6
t->3

as t approaches 3

so far i have:

-> 27 - 27 - 3 +3 / over 9 - 3 - 6

-> 0 / over 0

= 0

I substitute it 3 for t but i don't know if this is the right way for this one, Also i need to illustrate each step that involves the use of a limit property, but how? Do i have to label each step like by hypothesis, by substitution, by arithmetic etc..?

2. divide every term in the quotient by the highest power of x (that would be x^3).

now, what happens to the value of the expression as x gets large?

3. Originally Posted by Cyberman86
I can't figure out this one, its driving me crazy. My teacher made this problems up and he call them Interesting problems and sometimes even the tutors struggle with his problems.

Lim 5 + 2x^2 + x^3 / over 3 - x^2 + 2x^3
x->infinity

It has the sign of Infinity but i can't draw it and -> is an arrow as x approaches...

[snip]
There are numerous approaches. I'll use the simplest:

$\lim_{x \rightarrow + \infty} \frac{x^3 + 2x^2 + 5}{2x^3 - x^2 + 3}$

Divide numerator and denominator by $x^3$:

$= \lim_{x \rightarrow + \infty} \frac{1 + \frac{2}{x} + \frac{5}{x^3}}{2 - \frac{1}{x} + \frac{3}{x^3}} = \frac{1 + 0 + 0}{2 - 0 + 0} = \frac{1}{2}$.

4. Originally Posted by Cyberman86
[snip]
Evaluate the limit and illustrate each step that involves the use of a limit property.

Lim t^3 - 3t^2 - t + 3 / over t^2 - t - 6
t->3
as t approaches 3

so far i have:

-> 27 - 27 - 3 +3 / over 9 - 3 - 6

-> 0 / over 0

= 0

I substitute it 3 for t but i don't know if this is the right way for this one, Also i need to illustrate each step that involves the use of a limit property, but how? Do i have to label each step like by hypothesis, by substitution, by arithmetic etc..?
$\lim_{t \rightarrow 3} \frac{t^3 - 3t^2 - t + 3}{t^2 - t - 6} = \lim_{t \rightarrow 3} \frac{t^2 (t - 3) - (t - 3)}{(t-3)(t+2)}$.

Finish factorising the numerator, cancel the factor common to numerator and denominator and take the limit.

5. did i finish it correctly:

9 - (3 - 3) / over (3 + 2)

= 9/5

6. Originally Posted by Cyberman86
did i finish it correctly:

9 - (3 - 3) / over (3 + 2)

= 9/5
No.

Show all your working in factorising $t^2 (t - 3) - (t - 3)$. Then either someone can show you the mistake(s) you've made or you might actually realise the mistake yourself.

7. here i go again:

-> t^2 (t-3) - (t-3) / (t-3) (t+2)

-> numerator and denominator (t-3) cancel

-> t^2 - (t-3) / (t+2)

-> then i substitute t for 3?

-> 3^2 - (3-3) / (3+2)

-> 9 - (3-3) / (3+2)

-> 9 / 5

i showed all i did but still i don't know if that's what you mean

8. Originally Posted by Cyberman86
here i go again:

-> t^2 (t-3) - (t-3) / (t-3) (t+2)

-> numerator and denominator (t-3) cancel

-> t^2 - (t-3) / (t+2)

-> then i substitute t for 3?

-> 3^2 - (3-3) / (3+2)

-> 9 - (3-3) / (3+2)

-> 9 / 5

i showed all i did but still i don't know if that's what you mean
Learn to factorise properly and you'll probably have greater success with questions that depend on that skill.

$t^2 (t - 3) - (t - 3)$

Take out the common factor:

$= (t - 3) (t^2 - 1)$.