Determining a trigonometric limit

[Bunny-chan]

Homework Statement

Calculate the following limit:

Homework Equations

The Attempt at a Solution

I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression $(x+\pi)$ to $u$, but I wasn't very successful. To what kind of algebric device I could resort to? Or is there other way to deduce the limit?

 

[LCKurtz]

Homework Statement

Calculate the following limit:

View attachment 203337

Homework Equations

The Attempt at a Solution

I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression $(x+\pi)$ to $u$, but I wasn't very successful. To what kind of algebric device I could resort to? Or is there other way to deduce the limit?

The problem as written is continuous at $x=0$, so just plug it in. Or if both terms in the denominator are supposed to be cube roots, try L'Hospital's rule.

 

[Ray Vickson]

Homework Statement

Calculate the following limit:

View attachment 203337

Homework Equations

The Attempt at a Solution

I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression $(x+\pi)$ to $u$, but I wasn't very successful. To what kind of algebric device I could resort to? Or is there other way to deduce the limit?

I assume you meant to write $\sqrt[3]{x+\pi}$ instead of $3 \sqrt{x+\pi}$ in the denominator. If you do not want to (or are unable to) use calculus, use instead the algebraic identity $a^3-b^3 = (a-b)(a^2+a b + b^2)$ for appropriate $a$ and $b$.