Answer
(a) $5$, greater
(b) $3$, less
(c) $4.1$, greater
(d) $4+h$, yes; it is consistent with the above answers.
(e) $m=4$.
To find the tangent line at the given point, one can substitute the $"y"$ in $y=x^2$ with $"y=mx+d"$ and then force the resultant quadratic equation to have only one root (Please note that this method is not a general way to find tangent lines since there are many examples in which there exist lines passing through a point of the graph of a function while they are not tangent lines. However, if one applies this method and concludes the result which describes a unique line, then the method will be valid and the result will be correct).
The general form of equation of line is $y=mx+d$. This line passes through the point $(2,4)$, so $$4=m \cdot 2 +d \quad \Rightarrow \quad d= 4-2m \,.$$Now we substitute$"y"$ in $y=x^2$ with $"y=mx+4-2m"$, so we have $$mx+4-2m=x^2 \quad \Rightarrow \quad x^2-mx+2m-4=0$$If we want that the above quadratic equation has only one root, its discriminant must be zero. Thus,$$\Delta=b^2-4ac=m^2-8m+16=0 \quad \Rightarrow \quad (m-4)^2=0$$ $$\Rightarrow \quad m=4.$$
One may expect the answer because of the above limiting behavior, which comes from some numerical computations. But keep in mind that this way of reasoning is not a correct and rigorous way of mathematical reasoning. There exist many examples that such numerical computations may mislead.
Work Step by Step
To find the slope of a line joining two points $(x_1,y_1)$ and $(x_2,y_2)$, one can use the formula $m=\frac{y_2-y_1}{x_2-x_1}$.
The slope of the tangent line has been computed in the above answer (part (e)), and it is equal to $4$.
(a)$$m=\frac{9-4}{3-2}=5>4$$
(b)$$m=\frac{1-4}{1-2}=3<4$$
(c)$$m=\frac{4.41-4}{2.1-2}=4.1>4$$
(d)$$\frac{(2+h)^2-4}{(2+h)-2}=\frac{4+4h+h^2-4}{h}=\frac{4h+h^2}{h}=4+h$$
(i) for (a), $h=1 \quad \Rightarrow4+h=4+1=5 \quad \checkmark$
(ii) for (b), $h=-1 \quad \Rightarrow 4+h=4-1=3 \quad \checkmark$
(iii) for (c), $h=0.1 \quad \Rightarrow 4+h= 4+0.1=4.1 \quad \checkmark$
(e) It has been explained in the answer.
