# Math- Calculus

I need help with these problems, I cannot find a similar example to help me in the book:

1. Find lim x->infinity (e^(-2x) + sin x).
2. Find the derivative of sqrt(9-x) using the limit process.
3. Find lim x-> -infinity (x + sqrt(x^2 + 2x)).
4. Show that the equation e^x = 2+2x has a solution that is a negative number.

1. 1. done by just getting a "feel" for the numbers
e^(-2x) = 1/e^(2x)
so as x ---> infinitity, the denominator gets huge and the result approaches zero
as for the sinx it simply runs up and down between -1 and 1 no matter what x is or how large x is

So the lim(e^(-2x) + sinx) does not reach an actual value, but simply fluctuages between -1 and 1
e.g. x = 500000
value = e^(-1000000) + sin(50000)
= "almost" zero + .1778..
for x = 500002
I get "zero" + sin500002 = -.9688..

2.
dy/dx = lim( √(9-x) - √(9-x-h) )/h , as h --->0
= lim( √(9-x) - √(9-x-h) )/h * ( √(9-x) + √(9-x-h) )/( √(9-x) + √(9-x-h) )
= lim ( 9-x - (9-x-h)/(h(√(9-x) + √(9-x-h) )
= lim h/(h(√(9-x) + √(9-x-h) )
= lim 1/(√(9-x) + √(9-x-h) ) , as h ---> 0
= 1/(√(9-x) + √(9-x))
= 1/(2√(9-x) )

the x--->0 should be included in each line except the last two, I was just being lazy

3. since there is no denominator, nor are we taking √ of a negative, there should be no problem here,
Again, just get a feel for the numbers
look at the √(x^2 + 2x)
as gets larger into the negatives, x^2 makes it positive
and x^2 gets bigger much faster than 2x
so the x^2 eventually leaves the 2x behind and taking √ brings us back to x
so we have lim (-x + x) or lim 0, which is 0

4. e^x = 2+2x
let y1= e^x and let y2 = 2x+2, the latter is a straight line
let's graph both of these.
http://www.wolframalpha.com/input/?i=plot+y+%3D+e%5Ex+%2C+y%3D2x%2B2
As you can see, there are actually two solutions, one x is positive, the other is negative.
Let's actually solve the equation
http://www.wolframalpha.com/input/?i=solve+e%5Ex%3D2x%2B2
notice that x = -.768039 and x = +1.67835

testing:
LS = e^1.67835 = 5.35671
RS = 2(1.67835) + 2 = 5.3567 , close enough

posted by Reiny

First Name

## Similar Questions

1. ### Calculus

1.) Find the derivative of tan (sec x). 2.) Find the derivative if 1/x in four ways, using the limit process, power rule, quotient rule and implicit differentiation. 3.) Show that the derivative of sec^-1 x is 1/(|x|*sqrt(x^2
2. ### Calculus

Find the positive integers k for which lim ->0 sin(sin(x))/x^k exists, and then find the value the limit. (hint:consider first k=0, then k=1. Find the limit in these simple cases. Next take k=2 and finally consder k>2 and
3. ### Calculus

how do you find the limit at infinity of: lim(x->infinity) (x+2)/sqrt(64 x^2+1) Do you first change the square root on denominator to (64x^2+1)^-1/2 and then divide everything by zero. Please help me with this, I'm confused.
4. ### Calculus

I have two similar problems that I need help completing. Please show all your work. Question: Find the limit L. Then use the å-ä definition to prove the limit is L. 1. lim (2x+5) x->3 2. lim 3 x->6 Thank you for your

For what values of x is the graph of y = 8e^−x^2 concave down? (Enter your answer using interval notation.) I started by finding the second derivative and factoring and ended up getting 16e^-x^2 (2x^2-1) and I know up till
6. ### calculus - ratio test

infinity of the summation n=1: (e^n)/(n!) [using the ratio test] my work so far: = lim (n->infinity) | [(e^n+1)/((n+1)!)] / [(e^n)/(n!)] | = lim (n->infinity) | [(e^n+1)/((n+1)!)] * [(n!)/(e^n)] | = lim (n->infinity) |
7. ### calculus

A) How do you prove that if 0(<or=)x(<or=)10, then 0(<or=)sqrt(x+1)(<or=)10? B) So once that is found, then how can you prove that if 0(<or=)u(<or=)v(<or=)10, then
8. ### Check my CALCULUS work, please! :)

Question 1. lim h->0(sqrt 49+h-7)/h = 14 1/14*** 0 7 -1/7 Question 2. lim x->infinity(12+x-3x^2)/(x^2-4)= -3*** -2 0 2 3 Question 3. lim x->infinity (5x^3+x^7)/(e^x)= infinity*** 0 -1 3 Question 4. Given that: x 6.8 6.9
9. ### Derivative Calculus

I just need my answers checked. 1) Find the derivative. f(x)=3x^5/4 - 8/9x^9/8 + x^2 - 9x + 1 Answer: f’(x)= 15/4x^1/4 - x ^1/8 + 2x -9 2) Let f(x)= 7x^3 - 9x. Find f’(5) Answer: f’(x)=21x^2-9 f’(5)=516 3) Find the given
10. ### calc

find the area between the x-axis and the graph of the given function over the given interval: y = sqrt(9-x^2) over [-3,3] you need to do integration from -3 to 3. First you find the anti-derivative when you find the

More Similar Questions