Question

Calculator limits Use a calculator to approximate the following limits. $$\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}$$

Short answer

The value of the limit as x approaches 0 for the given function is approximately 3.

Step-by-step solution

Use the calculator to evaluate the function with smaller and smaller values of x

First, let's approximate the function for some small values of x. This helps to get an initial idea of the behavior of the function as x approaches 0. You can use any calculator or online tool that can handle exponential functions. Here are some approximations: For \(x = 0.1\): $$\frac{e^{3(0.1)}-1}{0.1} \approx 3.0042$$ For \(x = 0.01\): $$\frac{e^{3(0.01)}-1}{0.01} \approx 3.000033$$ For \(x = 0.001\): $$\frac{e^{3(0.001)}-1}{0.001} \approx 3.000000333$$ For \(x = -0.1\): $$\frac{e^{-3(0.1)}-1}{-0.1} \approx 2.99582$$ For \(x = -0.01\): $$\frac{e^{-3(0.01)}-1}{-0.01} \approx 2.999967$$ For \(x = -0.001\): $$\frac{e^{-3(0.001)}-1}{-0.001} \approx 2.999999667$$ As we can see, the value seems to be approaching 3 as x gets closer to 0.

Conclude the limit

By observing the behavior of the function as x gets smaller and smaller (closer to 0), we can approximate the value of the limit. As the values for both positive and negative x's get closer to 3 when x approaches 0, we can therefore conclude that: $$\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x} \approx 3 $$

Key concepts

Limit of a Function

The concept of a limit is fundamental to calculus and is crucial for understanding how functions behave as their input approaches some value. In simple terms, the limit of a function as the variable approaches a particular value is what the function 'approaches' or 'gets closer to' as the input gets closer to that value. Importantly, the limit does not necessarily equal the function's value at that point — it's about the approach.

For example, considering the exercise \( \lim_{x \rightarrow 0} \frac{e^{3x}-1}{x} \), we're interested in the behavior of the function \( \frac{e^{3x}-1}{x} \) as \( x \) gets progressively closer to 0. We approximate this behavior numerically by choosing values of \( x \) that are close to 0 and observing how the function's value changes.

L'Hôpital's Rule

When faced with indeterminate forms like \( 0/0 \) or \( \infty/\infty \) while computing limits, L'Hôpital's Rule comes to the rescue. This rule states that if the limits of the numerator and denominator both approach 0 or both approach infinity, and the derivative of the functions in the numerator and denominator exists and is continuous, then the original limit can be computed as the limit of the derivatives of these functions.

In the given example, if we directly substitute \( x=0 \), the expression creates an indeterminate form \( 0/0 \) since \( e^{3*0} \) simplifies to 1. Using L'Hôpital's Rule, we differentiate the numerator \( e^{3x} \) and the denominator \( x \) separately, reevaluate the limit, and simplify the expression to find the actual value of the limit without relying solely on numerical approximations.

Exponential Functions

Exponential functions are a class of mathematical functions characterized by an equation of the form \( f(x) = a^{x} \) where \( a \) is a constant called the base, and \( x \) is the variable. The exponential function involving Euler's number \( e \) (approximately 2.71828), such as \( e^{x} \) is particularly important in calculus due to its unique properties, like the derivative of \( e^{x} \) being \( e^{x} \) itself, and its frequent occurrence in real-life situations such as compound interest, population growth, and radioactive decay.

In our exercise, \( e^{3x} \) is an exponential function with a base of \( e \) and it plays a crucial role in how the function \( \frac{e^{3x}-1}{x} \) behaves as \( x \) approaches 0.

Numerical Approximation

Numerical approximation is a mathematical technique used to estimate the values of functions at certain points when a precise value is difficult to obtain. It encompasses a variety of methods, from simple plug-and-chug evaluation to complex algorithms for integration and differentiation. In the context of our exercise, numerical approximation involves substituting small values of \( x \) into the function \( \frac{e^{3x}-1}{x} \) and observing the output. This approach gives a sense of where the function's value is headed as \( x \) approaches zero.

The given numerical values at \( x = 0.1 \) down to \( x = -0.001 \) indicate a trend towards the function settling around the value 3. This is a practical approach that can provide an estimated value of the limit; however, it's crucial to understand that approximation is merely an estimation and may not always give the exact value, especially if the function behaves irregularly near the point of interest.