Do the graphs of the functions f have tangent lines at the...

Do the graphs of the functions $f$ have tangent lines at the given points? If yes, what is the tangent line?
$$f(x)=\left\{\begin{array}{ll}
\sqrt{x} & \text { if } x \geq 0 \\
-\sqrt{-x} & \text { if } x<0
\end{array} \text { at } x=0\right.$$

Key Concepts

Tangent Line

A tangent line is the line that just touches a curve at a given point and has the same instantaneous rate of change (derivative) at that point. In calculus, the tangent line is used to approximate the local behavior of a function near that point.

Differentiability

Differentiability refers to the existence of a finite derivative at a point, meaning the function has a well-defined, unique slope at that point. When evaluating differentiability, one typically checks that the left-hand and right-hand derivatives are equal and finite.

One-Sided Derivatives

One-sided derivatives are the limits of the difference quotient as one approaches the point of interest from the left or right. They are especially important for functions defined piecewise or at endpoints, as they reveal the behavior of the function from different directions.

Vertical Tangents

A vertical tangent occurs when the derivative tends to infinity or negative infinity as the point is approached, indicating that the tangent line is vertical. This is a special case where, despite the absence of a finite slope, a tangent line (vertical line) can still be defined.

Piecewise Functions

Piecewise functions are defined by different expressions on different parts of their domain. Analyzing such functions requires careful examination of their continuity and differentiability at the boundaries where the definition changes.

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