Is my conception of limit correct?

Question

$(1)$ Can I define the derivative $\left( \dfrac{dy}{dx}=\lim_{\Delta x \rightarrow 0}\dfrac{\Delta y}{\Delta x} \right)$ as a value which can never be reached when $\Delta x$ approaches zero but every value smaller (or greater in other cases) to it can be reached when $\Delta x$ approaches zero.

$(2) $Similarly can I define the definite integral $\displaystyle \left( \int^b_a y\ dx \right)$ as the value of Reimann sum (in which we take the smallest value of function on the interval $\Delta x$) which can never be reached when $\Delta x$ approaches zero but every value smaller to it can be reached when $\Delta x$ approaches zero.

Answer 1

No. Both are false. The value might be reached (trivial example for the derivative: a linear function). And it is also false that "every" value ( in what range?) is attained, as the same example shows.

Your assertion is ( more or less) right if applied to the values of the variable , instead of to the function.

Answer 2

No. Consider the example where $y=1$ for all values of $x$. Then $\frac{\Delta y}{\Delta x} = 0$ for all choices of $\Delta x\neq 0$ and this is also the limit as $\Delta x\to 0$. So the limit in this case is reached (contradicting your claim "can never be reached") and furthermore no value smaller or greater than the limit is ever reached (contradicting your claim "every value smaller or greater to it can be reached").

The right description is that $\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x} = L$ means that for any choice of a small $\varepsilon>0$ the values of $\frac{\Delta y}{\Delta x}$ are inside the range $(L-\varepsilon, L+\varepsilon)$ as long as $\Delta x$ is close enough to $0$. More precisely, for any $\varepsilon >0$ there exists $\delta > 0$ such that when $-\delta < \Delta x < \delta$, then $L-\varepsilon < \frac{\Delta y}{\Delta x} < L+\varepsilon$.