Question

Limit   

x~0       (√x+4)-2
              --------
                 X
Please solve it in detail

Answer 1

Rationalize the numerator:

`\frac{√(x+4)-2}x\cdot(√(x+4)+2)/(√(x+4)+2)=((√(x+4))^2-2^2)/(x(√(x+4)+2))=\frac x{x(√(x+4)+2)}=\frac1{√(x+4)+2}`

This is continuous at
`x=0`, so we can evaluate the limit directly by substitution:

`\displaystyle\lim_(x\to0)\frac{√(x+4)-2}x=\lim_(x\to0)\frac1{√(x+4)+2}=\frac1{\sqrt4+2}=\frac14`