Evaluate the Limit

Problem

(lim_x→8)((√(,x)+√(3,8*x)+4)/(√(,4*x)+√(3,x)+4))

Solution

1. Identify the limit expression and the value that x is approaching, which is x=8

2. Substitute the value x=8 into the numerator of the expression.

√(,8)+√(3,8*(8))+4

√(,4⋅2)+√(3,64)+4

2√(,2)+4+4

2√(,2)+8

3. Substitute the value x=8 into the denominator of the expression.

√(,4*(8))+√(3,8)+4

√(,32)+2+4

√(,16⋅2)+6

4√(,2)+6

4. Form the resulting fraction and simplify by factoring out common terms.

(2√(,2)+8)/(4√(,2)+6)

(2*(√(,2)+4))/(2*(2√(,2)+3))

(√(,2)+4)/(2√(,2)+3)

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 2√(,2)−3

((√(,2)+4)*(2√(,2)−3))/((2√(,2)+3)*(2√(,2)−3))

(2*(2)−3√(,2)+8√(,2)−12)/((2√(,2))2−3)

(4+5√(,2)−12)/(8−9)

(5√(,2)−8)/(−1)

8−5√(,2)

Final Answer

(lim_x→8)((√(,x)+√(3,8*x)+4)/(√(,4*x)+√(3,x)+4))=8−5√(,2)

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