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

Answer 1

#1#

#lim_(x->oo)[(x-cosx)/x]=lim_(x->oo)[x/x-cosx/x]# #=lim_(x->oo)[1-cosx/x]=lim_(x->oo)[1]-lim_(x->oo)[cosx/x]#
#->lim_(x->oo)[cosx/x]=0#
The range of #cosx# is #-1<=cosx<=1#, however as #x->oo# you are dividing a small number by an increasingly large number .
Thus: #=1-0=1#
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Answer 2

#lim_(x->∞ )(x-cosx)/x=1#

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

#lim_(x->∞ )(f(x)+g(x))=lim_(x->∞ )f(x)+lim_(x->∞ )g(x)#

Separate the terms and solve them individually

#lim_(x->∞ )(x-cosx)/x=lim_(x->∞ )x/x-lim_(x->∞)cosx/x#

The first term is

#lim_(x->∞ )x/x=∞/∞#

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

#lim_(x->∞ )f(x)/g(x)=lim_(x->∞ )f^'(x)/g^'(x)=L#

In this case

#lim_(x->∞ )x/x=lim_(x->∞ )1/1=1#

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

#g(x)<=f(x)<=h(x)# (1)

and

#lim_(x->∞)g(x)=lim_(x->∞)h(x)=L# (2)

then

#lim_(x->∞)f(x)=L# (3)

(1) Let
#f(x)=1/x*cosx#
#g(x)=-1/x#
#h(x)=1/x#

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

#-1<=cosx<=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 11/x and the minimum will be -11/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:

#-1/x<=1/x*cosx<=1/x#

This is saying that #1/x*cosx# is squeezed or sandwiched between #-1/x# and #1/x# as x approaches infinite. You can see this when f(x), g(x) and h(x) are plotted together.


From the graph, you can see that as x approaches ∞, f(x) is being squeezed toward 0 by #1/x# and #-1/x#. You can also see that #g(x)<=f(x)<=h(x)# , that is, f(x) is contained within the bounds of g(x) and h(x).

(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.

#lim_(x->∞)(1/x)=lim_(x->∞)(-1/x)=0=L#

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

#lim_(x->∞)cosx/x=L=0#

Finally, putting it all together

#lim_(x->∞ )(x-cosx)/x=lim_(x->∞ )x/x-lim_(x->∞)cosx/x=1-0=1#

You prove this for yourself by looking at the graph

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Answer 3

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.

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Answer from HIX Tutor

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.

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.

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