# Math Help - Need Help WIth some questions on limits

1. ## Need Help WIth some questions on limits

1) lim x->INFINITY
arctan(x^6 - x^8)
the answer is -pi/2, but i dont know how to get it

2) on pic (it will have a red number 2 on it)

3) on second pic (it will have a red number 3 on it)

for 2 and 3, both answers are on it, but i have no idea into how to get this
does anyone know?????????

2. Originally Posted by Sneaky
1) lim x->INFINITY
arctan(x^6 - x^8)
the answer is -pi/2, but i dont know how to get it

== Arctan is the inverse function of tan and tan x --> 00 if x --> Pi/2, and tan x --> -oo if x --> -Pi/2 (this is basic trigonometry, nothing fancy).
From here what you get.

2) on pic (it will have a red number 2 on it)

== Just apply the def. of derivative to the function f(x) = 2^x at the point x = 4.

3) on second pic (it will have a red number 3 on it)

in case the limt exists, we know f ' (0) = lim [f(x) - f(0)]/x when x --> 0, so:

lim [x sin(8/x) - 0]/x = lim sin(8/x) , and this doesn't exist since 8/x --> oo when x --> 0 from the right, and thus sin(8/x) swings between -1 and 1 without approaching any definite number (for example, choose x_n = 1/nPi so that 8/x_n = 8nPi, n an integer ==> sin(8/x_n) = 0 and clearly x_n --> 0 when n --> oo. Now choose y_n = 1/(n/16)Pi, with n an odd integer. again, y_n --> o when n-->oo, but this time sin(8/y_n) = (n/2)Pi, and since n is odd sin(8/y_n) is 1 or -1 ==> again, the limit doesn't exist.

Tonio

for 2 and 3, both answers are on it, but i have no idea into how to get this
does anyone know?????????
...............................

3. can you elaborate more on pic two, (the one with 2^x)
when i derive 2^x i get:
(ln2)( e^xln2)
when i sub in 4 i get:
16ln2

for evaluating the top limit
i am lost because there is a 2^x
i can simplify the bottum to get
(x-2)(x+2)
which will make the equation 0/12 -> 0

16ln2 does not equal 0
so i still dont see why this one is the answer....

4. Originally Posted by Sneaky
can you elaborate more on pic two, (the one with 2^x)
when i derive 2^x i get:
(ln2)( e^xln2)
when i sub in 4 i get:
16ln2

for evaluating the top limit
i am lost because there is a 2^x
i can simplify the bottum to get
(x-2)(x+2)
which will make the equation 0/12 -> 0

16ln2 does not equal 0
so i still dont see why this one is the answer....
What's the definition of a function's derivative?

5. derivitive is slope at a point x
isnt it the (f(x+h)-f(x))h^-1
when i use that i get:

2^(x+h) - 2^x
--------------
h

i dont know how to further simplify this...

6. Originally Posted by Sneaky
derivitive is slope at a point x
isnt it the (f(x+h)-f(x))h^-1
when i use that i get:

2^(x+h) - 2^x
--------------
h

i dont know how to further simplify this...
You do not have to differentiate anything. You're simply asked to identify the function and the value of $a$. This is easily done by comparing the expression with the definition $f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$. Surely you can make the comparison and identify what is $f(x)$ and what is $a$ ....