91Ó°ÊÓ

Understanding the continuity of functions is a fundamental aspect of calculus. For a function to be continuous at a point, it must satisfy three essential conditions:

• The function value at that point, \(f(a)\), must exist.
• The limit of the function as it approaches that point, \(\lim_{x \rightarrow a} f(x)\), must exist.
• The function value and the limit at that point must be equal, meaning \(\lim_{x \rightarrow a} f(x) = f(a)\).

Let's see how these criteria apply to the function \(f(x) = x^{1/3}\) at \(x = 0\). At \(x = 0\), \(f(0) = 0^{1/3} = 0\), so the first condition is satisfied. Then we calculate the limit of the function as \(x\) approaches \(0\), giving us \(\lim_{x \rightarrow 0} x^{1/3} = 0\). Since the limit equals the function value, \(f(x)\) is continuous at \(x = 0\). This continuity sets the stage for investigating vertical tangents.

Calculating the derivative of a function helps us understand its rate of change. For the function \(f(x) = x^{1/3}\), we apply the power rule for differentiation. The power rule states that if \(f(x) = x^n\), then \(f'(x) = n\cdot x^{n-1}\).
Using the power rule here gives us:\[f'(x) = \frac{d}{dx}(x^{1/3}) = \frac{1}{3} x^{-2/3}\]This derivative is a representation of the slope of the function at any given point other than \(x = 0\) since it involves a negative power of \(x\). Vertical tangents, however, require a deeper exploration of limits, which we will discuss next.

Limits play a crucial role in understanding vertical tangents, as they allow us to examine the behavior of a function or its derivatives as they approach particular points. In this context, we are interested in the limit of the absolute value of the derivative of \(f(x) = x^{1/3}\) as \(x\) approaches \(0\):\[\lim_{x \rightarrow 0} \left| \frac{1}{3} x^{-2/3}\right|\]To simplify, think of \(x^{-2/3}\) as \(\frac{1}{x^{2/3}}\). Substituting this, the limit becomes:\[\lim_{x \rightarrow 0} \left| \frac{1}{3x^{2/3}}\right| \]As \(x\) approaches \(0\), the denominator \(3x^{2/3}\) gets smaller and smaller, causing the entire fraction to grow larger towards infinity. This result, \(\infty\), suggests an infinite slope or a vertical tangent line at \(x = 0\). Thus, the equation of the vertical tangent line is \(x = 0\). This use of limits to identify vertical tangents is a vital technique in calculus.