Understanding Limits: How to Solve Challenging Problems in Calculus

[mohlam12]

hello everyone
we've been doing some exercices of limits at class and there are many ones that i didn't understand... and since you don't get that chance to ask your teacher after class in Morcco, I came here for help!
here are two of the tens that i didn't undersatnd:
so to solve this limit:

lim (x-1)/(sqrt(x²+1))
x-> +infinity

you have to go from or each x Є ]-infinity,0[ U ]0, +infinity[
g(x)=(x-1)/(sqrt(x²+1))
...x(1-1/x)
g(x)=-----------------
...|x| sqrt(1+1/x²)

i just want to understand how you go from that first line to th second line !?

and also on this one:

how to go from
$sqrt(x^2+x+1)-x$
to:
$x(sqrt(1+1/x+1/x^2)-1))$ for each x Є ]0,+infinity[

i really appreciate your help, and also if there is a website that gives you the trucks to solve these kind of limits...thanks again

 

[Galileo]

It's just bringing the x outside of the brackets. Isn't is clear that x(1-1/x) is the same as (x-1) for x/=0? Just expand the brackets.
Same thing with sqrt(x^2+1). You can bring out the x^2 in (x^2+1), giving x^2(1+1/x^2) (valid for x/=0)

But you don't need it to solve the limit. Intuitively you can argue that the -1 in the numerator and the +1 in the denominator are pretty insignificant for large x, so ignoring those you get x/|x|, whose limit is clear.
You can also simply divide top and bottom of the fraction by x.

 

[Architeuthis Dux]

Hello there,

It's great that you are seeking help in understanding limits in calculus. Limits can be a challenging concept to grasp, but with practice and a solid understanding of the principles, you can become proficient in solving them.

To solve the first limit (lim (x-1)/(sqrt(x²+1)) as x approaches positive infinity), we can use the concept of rationalizing the denominator. This means multiplying the expression by a cleverly chosen form of 1 to get rid of any square roots in the denominator. In this case, we can multiply by the expression (x+1)/(x+1), which is equal to 1, to get:

(x-1)/(sqrt(x²+1)) * (x+1)/(x+1)

Expanding this out, we get:

(x²-1)/(sqrt(x²+1) * (x+1)

Now, we can use the identity (a²-b²) = (a+b)(a-b) to rewrite the numerator as (x+1)(x-1). This gives us:

(x+1)(x-1)/(sqrt(x²+1) * (x+1)

The (x+1) term in the numerator and denominator cancel out, leaving us with:

(x-1)/sqrt(x²+1)

This is the same expression as the one in the second line of your problem. The rest of the steps involve simplifying the expression and taking the limit as x approaches infinity.

For the second limit (sqrt(x^2+x+1)-x), we can use a similar approach of rationalizing the denominator. In this case, we can multiply by the expression (sqrt(x^2+x+1)+x)/(sqrt(x^2+x+1)+x), which is equal to 1. This gives us:

(sqrt(x^2+x+1)-x) * (sqrt(x^2+x+1)+x)/(sqrt(x^2+x+1)+x)

Expanding this out, we get:

(x^2+x+1-x^2)/(sqrt(x^2+x+1)+x)

The (x^2-x^2) terms cancel out, leaving us with:

(x+1)/(sqrt(x^2+x+1)+x)

From here, we can use the same identity as before to rewrite the numerator as (x+1)(x-1). This gives us:

(x+1)(x-1)/(