To evaluate the limit

$\lim_{x \to 0} \frac{\sqrt[3]{1 + c \cdot x} - 1}{x},$

we can use the technique of L'Hôpital's rule or apply a series expansion approach. Here's a detailed evaluation using both approaches:

Method 1: L'Hôpital's Rule

L'Hôpital's Rule states that if a limit results in an indeterminate form like $\frac{0}{0}$, we can differentiate the numerator and denominator separately.

1. Identify the indeterminate form:

   As $x \to 0$, both the numerator and the denominator approach 0:

   $\frac{\sqrt[3]{1 + c \cdot x} - 1}{x} \to \frac{0}{0}.$

2. Differentiate the numerator and denominator:

   Differentiate the numerator $\sqrt[3]{1 + c \cdot x} - 1$ with respect to $x$:

   [ \text{Numerator derivative: } \frac{d}{dx} \left(\sqrt[3]{1 + c \cdot x} - 1\right) = \frac{d}{dx} \left(\sqrt[3]{1 + c \cdot x}\right). ]

   Using the chain rule,

   $\frac{d}{dx} \left(\sqrt[3]{1 + c \cdot x}\right) = \frac{1}{3} (1 + c \cdot x)^{-\frac{2}{3}} \cdot c = \frac{c}{3 (1 + c \cdot x)^{\frac{2}{3}}}.$

   Differentiate the denominator, which is $x$:

   $\text{Denominator derivative: } \frac{d}{dx} (x) = 1.$

3. Apply L'Hôpital's Rule:

   Substituting the derivatives into the limit:

   $\lim_{x \to 0} \frac{\frac{c}{3 (1 + c \cdot x)^{\frac{2}{3}}}}{1} = \frac{c}{3 (1 + c \cdot 0)^{\frac{2}{3}}} = \frac{c}{3}.$

Method 2: Series Expansion

We can also use a series expansion of the cube root function near $x = 0$.

1. Expand $\sqrt[3]{1 + c \cdot x}$ using the binomial approximation:

   For $|x|$ small,

   $\sqrt[3]{1 + c \cdot x} \approx 1 + \frac{c \cdot x}{3} + O(x^2).$

2. Substitute and simplify:

   Substitute this approximation into the original expression:

   $\frac{\sqrt[3]{1 + c \cdot x} - 1}{x} \approx \frac{\left(1 + \frac{c \cdot x}{3}\right) - 1}{x} = \frac{\frac{c \cdot x}{3}}{x} = \frac{c}{3}.$

Final Answer:

$\boxed{\frac{c}{3}}$

Would you like more details, or do you have any questions?

Related Questions:

1. How do you apply L'Hôpital's Rule for more complex limits?
2. What are other ways to evaluate limits involving indeterminate forms?
3. Can Taylor series be used to solve similar problems?
4. How does the chain rule work in differentiation?
5. What is the significance of binomial approximation in limit evaluations?

Tip:

For functions approaching indeterminate forms like $\frac{0}{0}$, L'Hôpital's Rule and series expansions are powerful tools that simplify evaluation.