Question

Discuss the continuity of the function (tan^-1 x)/x at x= 0

Answer 1

Answer:

Here's what I've done:

Let ϵ>0ϵ>0 be given.

Now,

|f(x)−f(y)|=|tan−1x−tan−1y|=∣∣tan−1(x−y1+xy)∣∣

|f(x)−f(y)|=|tan−1⁡x−tan−1⁡y|=|tan−1⁡(x−y1+xy)|

For non-negative x,y∈R,x,y∈R,

|f(x)−f(y)|=|tan−1x−tan−1y|=∣∣tan−1(x−y1+xy)∣∣≤|tan−1(x−y)|≤|x−y|<δ

|f(x)−f(y)|=|tan−1⁡x−tan−1⁡y|=|tan−1⁡(x−y1+xy)|≤|tan−1⁡(x−y)|≤|x−y|<δ

We can choose ϵ=δϵ=δ. Now, for negative x,y∈Rx,y∈R. Please, how do I go about it?