Evaluate the Limit

Problem

(lim_x→0)((√(,x+9)−3)/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in (√(,9)−3)/0=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x+9)+3

(lim_x→0)((√(,x+9)−3)/x)⋅(√(,x+9)+3)/(√(,x+9)+3)

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

(lim_x→0)(((√(,x+9))2−3)/(x*(√(,x+9)+3)))

4. Simplify the numerator by performing the squaring operations.

(lim_x→0)((x+9−9)/(x*(√(,x+9)+3)))

(lim_x→0)(x/(x*(√(,x+9)+3)))

5. Cancel the common factor of x from the numerator and the denominator, provided x≠0

(lim_x→0)(1/(√(,x+9)+3))

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

1/(√(,0+9)+3)

1/(3+3)

Final Answer

(lim_x→0)((√(,x+9)−3)/x)=1/6

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