Lim x -> Inf of (x + sinx)/(x + cosx) Approaches 1

  • Thread starter PrudensOptimus
  • Start date
In summary, the limit of (x + sinx)/(x + cos x) as x approaches infinity is equal to infinity. This is due to the fact that the function Sin[x]/x approaches zero as x approaches infinity and is bounded between -1/x and 1/x for x>0. As for the limit of (x+1)/12 as x approaches infinity, it is equal to infinity as well, because although the function is undefined at infinity, it approaches infinity from both positive and negative sides. Lastly, the limit of sinx/x as x approaches infinity is equal to zero, as the function is bounded and the reciprocal of 1/x approaches zero as x approaches infinity.
  • #1
PrudensOptimus
641
0
lim x-> inf, (x + sinx)/(x + cos x)

here's what i got:


(1 + (lim x ->inf, sinx/x))/(1 + (lim x->inf, cosx/x))

note the "1"s came from x/x from the original limit.

And lim x->inf, cosx/x becomes 0.
but I don't understand how lim x->inf, sinx/x becomes 0.
sinx/x = {1 - x^2/3! + x^4/5! - ... + ...} since x approaches infinity, that series would be 1 - inf + inf - inf, eventually it is still infinity.

so, we have (1 + inf)/(1 +0) = inf, but the text says it approaches 1, so it much be something wrong i did with the sinx/x thing.

Could someone please explain?
 
Mathematics news on Phys.org
  • #2
and another question about lim x->+-inf, (x+1)/12: how does it become +-inf? isn't it suppose to be No LImit because (x+1)/12 as x -> inf becomes 1+0/0??
 
  • #3
The limit is Sin[x]/x is more easily understood as zero if you apply the sandwich thereom.

-1 <= Sin[x] <= 1
-1/x <= Sin[x]/x <= 1/x, (x > 0)

Limit[ x->inf, -1/x ] = 0
Limit[ x->inf, 1/x ] = 0

So the function Sin[x]/x is always "squeezed" inbetween the functions 1/x and -1/x for x>0 by the second statement. Since these two functions themselves approach zero from above and below, Sin[x]/x, which is always between the two, is also forced go to zero.
 
  • #4
Originally posted by PrudensOptimus
and another question about lim x->+-inf, (x+1)/12: how does it become +-inf? isn't it suppose to be No LImit because (x+1)/12 as x -> inf becomes 1+0/0??

Well, by saying that the limit of a function as x approaches c is infinity really means: "this limit doesn't exist, but the closer you get to the number c, the larger the value of f(x) will get". It's a conveniant way of describing how the function misbehaves near the point c or towards infinity (or minus infinity).
 
  • #5
hat series would be 1 - inf + inf - inf, eventually it is still infinity.

This kind of series is an "indeterminate form", meaning the only thing it tells us is that we have to do more or different work.

The easiest way to see how the answer comes is to think back to trig and remember that (sin x) has a maximum value.
 
  • #6
Originally posted by Hurkyl
This kind of series is an "indeterminate form", meaning the only thing it tells us is that we have to do more or different work.

The easiest way to see how the answer comes is to think back to trig and remember that (sin x) has a maximum value.

because sin x isn't necessarily inf as x -> inf and the reciprocal of 1/x with x -> inf is naturally x -> zero why not check the limit of the reciprocal which gives x/sin x, as x -> zero, which has a limit of one.
Cheers, Jim
 
  • #7
To see the answer look at what happens when x gets large, and the difference between x and sin(x).

e.g. when x=100000000, |sin(x)|<= 1
 
  • #8
Another good way to find the limit value of sinx/x as x tends to inf is to use the following property.
Let f(x) and g(x) be two functions defined on an interval containing a, possible except a.
If f(x) is bounded and lim x->a h(x) = 0,
then lim x->a h(x)f(x)=0

lim sinx / x
x->[oo]

notice that 1/x tends to zero while |sinx|<=1 which is bounded, as x tends to inf.

Therefore the limit tends to zero.
 

1. What does the notation "Lim x -> Inf" mean?

The notation "Lim x -> Inf" represents the limit as x approaches infinity. This means that we are looking at the behavior of the function as x gets larger and larger.

2. How do I determine the limit of a function as x approaches infinity?

To determine the limit of a function as x approaches infinity, we can evaluate the function at larger and larger values of x. If the function approaches a specific value or approaches infinity, then that value is the limit. In this case, as x gets larger and larger, the function (x + sinx)/(x + cosx) approaches 1.

3. Why is the limit of (x + sinx)/(x + cosx) equal to 1 as x approaches infinity?

This limit can be determined using the quotient rule for limits, which states that if the limit of f(x) and g(x) both exist and g(x) does not equal 0, then the limit of f(x)/g(x) is equal to the limit of f(x) divided by the limit of g(x). In this case, as x approaches infinity, both the numerator and denominator of (x + sinx)/(x + cosx) approach infinity, so we can use the quotient rule to determine that the limit is equal to the limit of x/x, which is equal to 1.

4. Is there a graphical representation of the limit of (x + sinx)/(x + cosx) as x approaches infinity?

Yes, the limit can be graphically represented by a horizontal asymptote. As x approaches infinity, the graph of (x + sinx)/(x + cosx) gets closer and closer to the horizontal line y = 1. This means that the limit of the function is equal to 1.

5. How does the value of the limit of (x + sinx)/(x + cosx) change as x approaches infinity?

As x approaches infinity, the value of the limit of (x + sinx)/(x + cosx) remains constant at 1. This means that no matter how large x gets, the function will approach 1, indicating a stable behavior of the function at infinity.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
124
  • General Math
Replies
3
Views
882
  • Calculus and Beyond Homework Help
Replies
3
Views
268
Replies
4
Views
8K
  • Calculus and Beyond Homework Help
Replies
5
Views
979
  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
Replies
4
Views
281
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • General Math
Replies
1
Views
2K
Replies
2
Views
1K
Back
Top