# #lim_(x->0)sin(1/x)/(sin(1/x))# ?

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Find the limit #lim_(x->0)sin(1/x)/(sin(1/x))#

How would you approach this? Is it #1# or it doesn't exist?

Find the limit

How would you approach this? Is it

##### 2 Answers

# lim_(x rarr 0) \ sin(1/x)/(sin(1/x)) = 1 #

#### Explanation:

we seek:

# L = lim_(x rarr 0) \ sin(1/x)/(sin(1/x)) #

When we evaluate a limit we look at the behaviour of the function "near" the point, not necessarily the behaviour of the function "at" the point in question, thus as

# L = lim_(x rarr 0) \ sin(1/x)/(sin(1/x)) #

# \ \ = lim_(x rarr 0) \ 1 #

# \ \ = 1 #

For clarity a graph of the function to visualise the behaviour around

graph{sin(1/x)/sin(1/x) [-10, 10, -5, 5]}

It should be made clear that the function

Please see below.

#### Explanation:

The definitions of limit of a function I use are equivalent to:

Because of the meaning of "

That is, for the required

All of this the gets us:

(

Therefore,

**A nearly trivial example**