Question

What is the limit as `x` approaches 0 of `tanx/x`?

Answer 1

1

Explanation:

`lim_(x->0)tanx/x`

graph{(tanx)/x [-20.27, 20.28, -10.14, 10.13]}

From the graph, you can see that as `x->0`, `tanx/x` approaches 1

Answer 2

Remember the famous limit:

`lim_(x->0) sinx/x = 1`

Now, let's look at our problem and manipulate it a bit:

`lim_(x->0) tanx/x`

`= lim_(x->0) (sinx "/" cosx)/x`

`= lim_(x->0) ((sinx/x)) / (cosx)`

`= lim_(x->0) (sinx/x) * (1/cosx)`

Remember that the limit of a product is the product of the limits, if both limits are defined.

`= (lim_(x->0)sinx/x) * (lim_(x->0)1/cosx)`

`= 1 * 1/cos0`

`= 1`

Final Answer