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

Question

154061 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-12T13:13:31

1

#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 · 2018-06-12T13:24:05

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