Answer:
First, we need to find mtan using the given formula:
mtan = lim h→0 [f(a+h) - f(a)] / h
Plugging in a = 3 and f(x) = √(√3x + 7), we get:
mtan = lim h→0 [√(√3(3+h) + 7) - √(√3(3) + 7)] / h
Simplifying under the square roots:
mtan = lim h→0 [√(3√3 + √3h + 7) - 4] / h
Multiplying by the conjugate of the numerator:
mtan = lim h→0 [(√(3√3 + √3h + 7) - 4) * (√(3√3 + √3h + 7) + 4)] / (h * (√(3√3 + √3h + 7) + 4))
Using the difference of squares:
mtan = lim h→0 [(3√3 + √3h + 7) - 16] / (h * (√(3√3 + √3h + 7) + 4))
Simplifying the numerator:
mtan = lim h→0 [(√3h - 9) / (h * (√(3√3 + √3h + 7) + 4))]
Using L'Hopital's rule:
mtan = lim h→0 [(√3) / (√(3√3 + √3h + 7) + 4)]
Plugging in h = 0:
mtan = (√3) / (√(3√3 + 7) + 4)
Now we can use this to find the equation of the tangent line at P(3,4):
m = mtan = (√3) / (√(3√3 + 7) + 4)
Using the point-slope form of a line:
y - 4 = m(x - 3)
Simplifying and putting in slope-intercept form:
y = (√3)x/ (√(3√3 + 7) + 4) - (√3)9/ (√(3√3 + 7) + 4) + 4
This is the equation of the tangent line at P.