What is the slope of the line through (2,4) and (x,y) for y = x2 + x - 2 and x=1.99? x=2.004? x=2+h. What happens to this last slope when h is very small?

Answer:Part 1) [tex]m=4.99[/tex]Part 2) [tex]m=5.004[/tex]Part 3) [tex]m=h+5[/tex]Part 4) The slope tends to 5.Step-by-step explanation:we know thatThe formula to calculate the slope between two points is equal to [tex]m=\frac{y2-y1}{x2-x1}[/tex] we havethe points (2,4) and (x,y)[tex]y=x^{2} +x-2[/tex]Part 1)For x=1.99substitute the value of x in the quadratic equation and solve for y[tex]y=(1.99)^{2} +(1.99)-2[/tex][tex]y=3.9501[/tex]we have the points (1.99,3.9501) and (2,4) substitute the values in the formula of slope[tex]m=\frac{4-3.9501}{2-1.99}[/tex] [tex]m=\frac{0.0499}{0.01}[/tex] [tex]m=4.99[/tex]Part 2)For x=2.004substitute the value of x in the quadratic equation and solve for y[tex]y=(2.004)^{2} +(2.004)-2[/tex][tex]y=4.020016[/tex]we have the points  (2,4) and (2.004,4.020016)substitute the values in the formula of slope[tex]m=\frac{4.020016-4}{2.004-2}[/tex] [tex]m=\frac{0.020016}{0.004}[/tex] [tex]m=5.004[/tex]Part 3)For x=2+hsubstitute the value of x in the quadratic equation and solve for y[tex]y=(2+h)^{2} +(2+h)-2\\\\y=4+4h+h^{2} +2+h-2\\\\y=h^{2}+5h+4[/tex]we have the points(2,4) and (2+h,h^{2}+5h+4)substitute the values in the formula of slope[tex]m=\frac{h^{2}+5h+4-4}{2+h-2}[/tex] [tex]m=\frac{h^{2}+5h}{h}[/tex] Simplify[tex]m=h+5[/tex]Part 4) What happens to this last slope when h is very small[tex]m=h+5[/tex]If the value of h is very small, then the value of h tends to zero and the value of m tends to 5.therefore[tex]m=5[/tex]