# How do you find the limit of #(x-cosx)/x# as x approaches #oo#?

By signing up, you agree to our Terms of Service and Privacy Policy

Firstly, the limit of a sum is the sum of the limits

Separate the terms and solve them individually

The first term is

Use L'Hôpital's Rule for such limits of indeterminate form

In this case

Use the Squeeze or Sandwich Theorem for the second term. This is more complicated and involves three steps. We will apply them in order.

If

and

then

(1) Let

We know that cos(x) goes from -1 to 1

For f(x), we are multiplying 1/x by cos(x), which means multiplying by numbers from -1 to 1.

The upper and lower bounds of f(x) can now be found. The maximum value the function can take is 1*1/x and the minimum will be -1*1/x. Multiplying 1/x by any other number in-between -1 and 1 will result in a smaller number within these bounds. This means that the following is true:

This is saying that

From the graph, you can see that as x approaches ∞, f(x) is being squeezed toward 0 by

(2) Although I just gave the answer, pretend that you didn't have that graph and move on to step two. We need to find the limits of g(x) and h(x) and see if they are equal.

(3) They are equal, so from the Squeeze Theorem

Finally, putting it all together

You prove this for yourself by looking at the graph

By signing up, you agree to our Terms of Service and Privacy Policy

To find the limit of (x - cos(x))/x as x approaches infinity, we can use the concept of L'Hôpital's Rule. By applying this rule, we differentiate the numerator and denominator separately with respect to x.

Differentiating the numerator (x - cos(x)) with respect to x gives us 1 + sin(x), and differentiating the denominator (x) with respect to x gives us 1.

Now, we can evaluate the limit of (1 + sin(x))/1 as x approaches infinity. Since sin(x) oscillates between -1 and 1, the limit does not exist.

Therefore, the limit of (x - cos(x))/x as x approaches infinity is undefined.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the limit of #2x+5# as #x->-oo#?
- How do you find the limit of # (X^2 – x – 6) / (X^2 – 5x + 6)# as x approaches 0?
- How do you evaluate the limit #lim_(x to -1) (3x)/(x^2+2x+1)#?
- How do you find the limit of #cscx-cotx+cosx# as x approaches 0?
- How do you find the limit of # (x^2 +6) / (6x^3 +x^2 -1)# as x approaches #-oo#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7