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3 votes
Limit   

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

Limit    x~0 (√x+4)-2 -------- X Please solve it in detail-example-1

1 Answer

7 votes

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

User Steven Liao
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