Lim x3 3-x\3-√x+6

To find the limit of the given expression, we need to evaluate it as x approaches a certain value. In this case, we need to find the limit as x approaches 6.

lim x→6 (x^3 + 3 - x)/(3 - √x + 6)

To evaluate this limit, we can try direct substitution:

lim x→6 (6^3 + 3 - 6)/(3 - √6 + 6)

= (216 + 3 - 6)/(3 - √6 + 6)

= 213/√6 + 3

Since direct substitution didn't lead to a defined value, we need to use algebraic manipulation or another method to evaluate the limit.

One possible method is to multiply the numerator and denominator by the conjugate of the denominator to eliminate the square root:

lim x→6 [(x^3 + 3 - x)(3 + √x - 6)]/[(3 - √x + 6)(3 + √x - 6)]

= [(x^3 + 3 - x)(3 + √x - 6)]/[9 - x]

= [x^4 - 5x^3 - 2x^2 + 19x - 15]/[9 - x]

Now we can try direct substitution again:

lim x→6 [6^4 - 5(6)^3 - 2(6)^2 + 19(6) - 15]/[9 - 6]

= [1296 - 1080 - 72 + 114 - 15]/3

= 261/3

= 87

So, the limit of the expression as x approaches 6 is 87.