Given function f(x)=√x. Find the formula for the slope of the secant line crossing the graph at the points x=4, and x=4+h.?

(a)
m
= [ f(4+h) - f(4) ] / [ (4+h) - 4 ]
= [ √(4+h) - √4 ] / h
= [ √(4+h) - 2 ] / h ..... Ans

(b)
as h → 0
[ √(4+h) - 2 ] / h
= [ √(4+h) - 2 ]*[ √(4+h) + 2 ] / { h*[ √(4+h) + 2 ] }
= [ (4+h) - 4 ] / { h*[ √(4+h) + 2 ] }
= h / { h*[ √(4+h) + 2 ] }
→ 1 / [ √(4+h) + 2 ] , because h → 0 implies h ≠ 0
→ 1/4 ..... Ans

(c)
the eq. of the tangent line is
y - f(4) = (1/4)( x - 4 )
y - 2 = (1/4)x - 1
y = (1/4)x + 1 ..... Ans