Question

limit as x approaches pi of sin(x+sinx)

Answer 1

just plug in x= pi in your function

Answer 2

sorry im trying to use continuity to evaluate the limit

Answer 3

You mean you want to use the definition? Or just verify if it is continuous at that point?

Answer 4

verifying that it is continuous

Answer 5

a function is said to be continuous at a point if the following relation holds:
\[\large \lim_{x \to x_0} f(x)=f(x_0) \]

Answer 6

awesome thanks

Answer 7

For this exercise you might want to reason (I assume this has been proven to you already) that \(\sin\) is continuous, so you can interchange the limit with the function. You can also reason that the above function is continuous because it is a composition of continuous functions.

Answer 8

well i was thinking that it was continuous at every number because of sin

Answer 9

it is, the above definition I have given you is read as continuity at a given point \(x=x_0\) to show that sin is continuous everywhere is a bit more complicated, but there is a lot of material for that online already.

What I have written above is nothing but the \(\epsilon, \delta\) criteria for continuity written in the form of a limit.