## Derivative Does Not Exist at a Point

The derivative at a point exists when the limit at the point exists:

Therefore, **if the limit doesn’t exist, then the derivative doesn’t exist either.** There are two general situations where this might happen:

- When the curve doesn’t have a tangent line because of a discontinuity or sharp corner.
- When the tangent line has an infinite slope (i.e. there is a vertical inflection point).

## 7 Examples of When the Derivative Does Not Exist at a Point

There are a **few specific situations** that causes a curve not to have a tangent line or an infinite slope:

**Jump discontinuity**: a gap in a graph means that the function is not continuous and therefore not differentiable.**Hole in the graph**: Holes (more formally called removable discontinuities) are tiny gaps in a graph. One example of when a hole happens is when a rational function has an x-value, that when plugged into the function causes both the numerator and the denominator to equal 0.

**Unbounded behavior**or an infinite discontinuity. The x-values get larger and larger as you try and move towards the point in question.**Sharp points**or cusps make a function not differentiable at that point.- The
**function can’t be defined**. For example, the square root function is not defined for values less than zero, so the derivative does not exist at any point less than zero. - The function can be defined but
**the derivative is infinity**at the point in question (or it doesn’t exist at all). Derivatives equal to infinity happens quite often with rational functions and it means that there is a vertical tangent at that point **Oscillating behavior.**Some functions behave badly and have oscillating discontinuities near certain points.

## References

Hole image created with Desmos.com.

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