I don’t really understand why hyp approaches delta(x) and angle Phi approaches x here, as delta(x) gets smaller and smaller. Can anyone explain, please? Thank you!
Hello @LinhPhan,
It seems the proofs are not covered by the course, but if you are interested, I am giving you some hints but I will leave the burden of algebra to you.
For the hypotenuse, use the laws of cosines (for ΔACD) and the taylor expansion for cos(x). You may want to google the names for what they are.
For the angle phi, if you look at my graph above, there are three unknown angles alpha, beta and phi, but you can also use them to write down three relations out of two triangles ΔABC and ΔACD and solve the angles in terms of x and Δx.
If needed and you show me your work, we can discuss it 
Cheers,
Raymond