14. \( \lim _{\Delta \rightarrow} \frac{(x+\Delta x)^{2}-x^{2}}{\Delta x} \)

To evaluate the limit \[ \lim_{\Delta x \to 0} \frac{(x+\Delta x)^{2} - x^{2}}{\Delta x}, \] we start by simplifying the expression in the numerator. We can expand \((x + \Delta x)^2\): \[ (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2. \] Now, substituting this back into the limit gives us: \[ \lim_{\Delta x \to 0} \frac{(x^2 + 2x\Delta x + (\Delta x)^2) - x^2}{\Delta x}. \] The \(x^2\) terms cancel out: \[ \lim_{\Delta x \to 0} \frac{2x\Delta x + (\Delta x)^2}{\Delta x}. \] Next, we can factor out \(\Delta x\) from the numerator: \[ \lim_{\Delta x \to 0} \frac{\Delta x(2x + \Delta x)}{\Delta x}. \] Assuming \(\Delta x \neq 0\), we can cancel \(\Delta x\): \[ \lim_{\Delta x \to 0} (2x + \Delta x). \] Now, we can directly substitute \(\Delta x = 0\): \[ 2x + 0 = 2x. \] Thus, the limit evaluates to: \[ \boxed{2x}. \]