Question

how to calculate the limit:

*I got confused especially with the square root.



1. lim as x approaches 2 (squareroot x^2 +5)-3 / (x-2)


2. lim as θ (theta) approaches zero (tan^2 (5θ) / (sin (3θ) sin (2θ) )

Answer 1

[ 2 sqrt(x^2+5) -3 ] / (x-2)

let x = 2+h then x^2 = 4 + 4h + h^2
and x^2 + 5 = h^2 + 4 h + 9

[ 2 sqrt (h^2 + 4h + 9) -3 ] /(2 +h -2)

[ 2 sqrt (9 + 4h + h^2) -3 ] / h

as h ---> 0 that looks like 6/h
which is undefined. I wonder if a typo

2. as θ -->0
tan^2 θ ---> (sin θ /cos θ)^2 -->θ^2
because sin θ --->θ and cos θ --> 1
so 25 θ^2 on the top
on the bottom you have
3θ * 2θ = 6 θ^2
so 25/6
θ^2