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In the definition of derivation, why $\Delta x$ approach to zero? I searched, but couldn't find a convincing answer.

user577215664
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AK Math
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    What level of understanding do you have about the derivative function? – Andrew Chin Mar 10 '22 at 20:57
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    The derivative tells us the slope of the tangent line at any point. We calculate the slope of the tangent line as the limit of the slope of secant lines. The secant lines approaches the tangent line as $\Delta x \rightarrow 0$ –  Mar 10 '22 at 21:00
  • $\Delta x$ is a measure of the error in your approximation. You want it to go to zero because you would like to have no error. – CyclotomicField Mar 10 '22 at 21:17

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As Frog Will Do mentions, you take that limit because the slope of the tangent line is the slope of the secant line between points that get arbitrarily close together.

There are alternatives though without limits. A tangent line to a point on, say, a parabola, can be constructed using algebra. Likewise a tangent circle: Derivatives without limits

TurlocTheRed
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