Finding Limit Algebraically?

Question

Limit of [root of (h^2 + 4h + 5) - root of (5)] / h when h approaches to 0 ^ (-)

Answer 1

[h→0⁻]lim (√(h²+4h+5)-√5)/h
[h→0⁻]lim (h²+4h+5-5)/(h(√(h²+4h+5)+√5))
[h→0⁻]lim (h²+4h)/(h(√(h²+4h+5)+√5))
[h→0⁻]lim (h+4)/(√(h²+4h+5)+√5)
4/(√5+√5)
2/√5

Answer 2

Limit of [root of (h^2 + 4h + 5) - root of (5)] / h when h approaches to 0 ^ (-)


You need to create something to get rid of the sq roots, so we create the same, but with he opposite sign = [root of (h^2 + 4h + 5) + root of (5)]. We need to put the same thing in the numerator and denominator. This way it would be he same as multipling by 1. By doing this, in the numerator we have a difference of squares that has the form of (a + b) * (a-b) = a^2 - b^2

{[root of (h^2 + 4h + 5) - root of (5)] * root of (h^2 + 4h + 5) + root of (5)] / h} / h * [root of (h^2 + 4h + 5) + root of (5)]

Now the numerator would be

h^2 + 4h + 5 - 5 / h * [root of (h^2 + 4h + 5) + root of (5)]

We do common factor in the numerator

h * ( 4 + h) / h * [root of (h^2 + 4h + 5) + root of (5)]

We simplify the h.

( 4 + h) / [root of (h^2 + 4h + 5) + root of (5)] and now we do the liimit.

( 4 + 0) / root of 0^2 + 4*0 + 5 + root of 5

4 / 2 * root of 5 = 2 / root of 5

That's the limit