Evaluate the Limit

Problem

(lim_x→0)((√(,x+4)−2)/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in (√(,4)−2)/0=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x+4)+2

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

(√(,x+4)−2)*(√(,x+4)+2)=(x+4)−4

4. Simplify the numerator to x and rewrite the limit expression.

(lim_x→0)(x/(x*(√(,x+4)+2)))

5. Cancel the common factor of x from the numerator and the denominator.

(lim_x→0)(1/(√(,x+4)+2))

6. Evaluate the limit by substituting x=0 into the simplified expression.

1/(√(,0+4)+2)=1/(2+2)

Final Answer

(lim_x→0)((√(,x+4)−2)/x)=1/4

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