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Magazine Chapter 3: Problem 72
Find the limit by interpreting the expression as an appropriate derivative. $$ \lim _{x \rightarrow 0} \frac{\exp \left(x^{2}\right)-1}{x} $$
Short Answer
Expert verified
The limit is 0.
Step by step solution
01
Identify the Derivative Form
Recognize that the given limit expression can be interpreted as a derivative at a point. The expression \[\lim _{x \rightarrow 0} \frac{\exp \left(x^{2}\right)-1}{x}\] is similar to the form \[f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}\]and thus can be seen as the derivative of a function evaluated at a specific point.
02
Identify the Function and Its Derivative
Identify the function to be differentiated. Here, let鈥檚 consider \( f(x) = e^{x^2} \). We need to find the derivative \( f'(x) \) at \( x = 0 \). The function \( f(x) = e^{g(x)} \) requires the chain rule for differentiation.
03
Apply Chain Rule to Find Derivative
Using the chain rule:\[ f(x) = \exp(g(x)) \quad \text{where} \quad g(x) = x^2\]The derivative of \( f(x) \) is:\[ f'(x) = \frac{d}{dx}\left( \exp(g(x)) \right) = \exp(g(x)) \cdot g'(x)\]where \( g'(x) = \frac{d}{dx}(x^2) = 2x \). Thus, \[ f'(x) = \exp(x^2) \cdot 2x.\]
04
Evaluate the Derivative at the Point
Evaluate \( f'(x) \) at \( x = 0 \): \[ f'(0) = \exp(0^2) \cdot 2(0) = 1 \cdot 0 = 0.\]
05
Conclude the Limit Value
Based on the steps above, we have determined that evaluating the derivative of the function \( f(x) = e^{x^2} \) at the point \( x = 0 \) yields a result of \( 0 \). Therefore, the limit of the original expression is:\[ \lim _{x \rightarrow 0} \frac{\exp \left(x^{2}\right)-1}{x} = f'(0) = 0.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
The concept of a derivative is crucial in understanding how the rate of change occurs in functions. Simply put, the derivative represents the slope of a function at any given point. Imagine the curve of a function on a graph. The derivative looks at how this curve rises or falls as you inch along it.
- A derivative tells us the instantaneous rate of change. Think of it like speed in a car; it shows how the speedometer changes as you press the accelerator.
- Mathematically, it鈥檚 defined by the limit process: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). The "h" represents an incredibly small change in "x."
- Interpreting this for any function gives us what we call the 'derivative.' This derivative can be used for various real-world applications, like understanding growth rates or predicting trends.
Chain rule
The chain rule is a critical tool for differentiating composite functions, where one function is nested inside another. Imagine having a set of Russian dolls. Each doll is inside another, just as a composite function is one function inside another.
- In differentiation, the chain rule allows us to "untangle" these nested functions.
- The formula is straightforward but powerful: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \).
- This rule lets us differentiate complex functions by first focusing on the "outside" function, then addressing the "inside" function.
Exponential function
The exponential function, often written as \( e^x \), forms the backbone of growth processes in mathematics. This function can model phenomena such as population growth, investment returns, or even radioactive decay.
- Exponential functions are special because their rate of change increases proportionally to the value of the function. A larger function value means a larger change rate.
- Mathematically, the derivative of the exponential function \( e^x \) is unique because it remains \( e^x \). This leads to simple yet profound rules of growth and change.
- Together with logarithms, exponential functions also help solve differential equations and are pivotal in natural growth models.
