Evaluate limit.

Question

As x approaches √2 find limit √(3+2x) - (√2+1) /( x^(2) -2 )

Answer 1

 

(√(3+2x)-(√2+1))/(x²-2) can be rewritten by multiplying top and bottom by:

√(3+2x)+(√2+1).

This changes the numerator to:

3+2x-(√2+1)²=3+2x-3-2√2=2(x-√2).

The denominator factorises: x²-2=(x-√2)(x+√2).

Both expressions have x-√2 as a common factor.

So, the original expression changes to:

2/((x+√2)(√(3+2x)+(√2+1)).

The common factor happens to take the value 0 when x=√2, therefore x→√2 will approach the above expression when x=√2.

This value is:

2/((√2+√2)(√(3+2√2)+(√2+1))=0.14645 approx.

So this is the limit as x→√2.