Lim┬(x→0)⁡〖(√(1+x+x^2 )-1)/tan⁡5x 〗

Evaluting the limand directly at x = 0 yields the indeterminate form 0/0, so we have a candidate for L'Hopital's rule. We get

If you don't know about L'Hopital's rule, or are otherwise not allowed to use it, you can instead rely on algebraic manipulation and a well-known limit,

where a ≠ 0. We write

The limit of a product is equal to the product of limits:

The first limit is 1 and the second limit is 1/5. For the remaining limit, multiply through the fraction by the conjugate of the numerator:

The remaining limand is continuous at x = 0, and its limit is

and hence the original limit is again 1 × 1/5 × 1/2 = 1/10.

1. perpendicular

2. parallel

3. perpendicular