Problem 16
Question
Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{x} $$
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: Identify the form of the limit
First, observe the given limit \( \lim _{x \rightarrow 0} \frac{e^{x}-1}{x} \). Plugging in \( x = 0 \), the expression becomes \( \frac{e^0 - 1}{0} = \frac{0}{0} \), which is an indeterminate form.
2Step 2: Use L'Hôpital's Rule
Since the limit is in an indeterminate form \( \frac{0}{0} \), we can use L'Hôpital's Rule, which allows us to differentiate the numerator and the denominator separately.
3Step 3: Differentiate the numerator and denominator
Differentiate the numerator \( e^x - 1 \) to get \( e^x \) and the denominator \( x \) to get \( 1 \).
4Step 4: Apply L'Hôpital's Rule
The limit now simplifies to \( \lim _{x \rightarrow 0} \frac{e^{x}}{1} = \lim _{x \rightarrow 0} e^{x} \).
5Step 5: Evaluate the simplified limit
Substituting \( x = 0 \) into \( e^x \), we find that \( e^0 = 1 \). Thus, the limit evaluates to 1.
Key Concepts
L'Hôpital's ruleIndeterminate formDifferentiationExponential function
L'Hôpital's rule
L'Hôpital's rule is a powerful tool in calculus for finding limits that initially appear to be in an indeterminate form, like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It can convert these forms into determinate ones, which allows us to evaluate the limits directly with differentiation.
When you encounter such forms while trying to evaluate a limit, L'Hôpital's rule states that if \( \lim_{x \to a} f(x) = 0 \) and \( \lim_{x \to a} g(x) = 0 \), or both limits result in \( \pm \infty \), then you can compute:
It is important to remember only to apply L'Hôpital's rule when the limit is in an indeterminate form! Differentiation might not help if not used in this correct context.
When you encounter such forms while trying to evaluate a limit, L'Hôpital's rule states that if \( \lim_{x \to a} f(x) = 0 \) and \( \lim_{x \to a} g(x) = 0 \), or both limits result in \( \pm \infty \), then you can compute:
- \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \)
It is important to remember only to apply L'Hôpital's rule when the limit is in an indeterminate form! Differentiation might not help if not used in this correct context.
Indeterminate form
An indeterminate form in the context of limits indicates a form that doesn’t directly resolve to a numerical value upon substitution. The form \( \frac{0}{0} \) that we encountered in the original exercise is one of the classic indeterminate forms.
Indeterminate forms occur because both the numerator and denominator tend to zero, leaving uncertainty about which goes to zero "faster". This is where the true value of the limit is hidden.
Other common indeterminate forms include:
Indeterminate forms occur because both the numerator and denominator tend to zero, leaving uncertainty about which goes to zero "faster". This is where the true value of the limit is hidden.
Other common indeterminate forms include:
- \( \frac{\infty}{\infty} \)
- \( 0 \times \infty \)
- \( \infty - \infty \)
- \( 0^0 \)
- \( 1^\infty \)
- \( \infty^0 \)
Differentiation
Differentiation is the process of finding the derivative. In this context, it gives us a tool to transform limits in indeterminate forms into ones that can be easily calculated using L'Hôpital's rule.
The derivative of a function indicates its rate of change. For the exponential function \( e^x \) that appeared in the exercise, the derivative is remarkably straightforward because it remains \( e^x \). It makes exponential functions convenient to work with in calculus.
To differentiate effectively, you’ll often rely on some basic rules:
The derivative of a function indicates its rate of change. For the exponential function \( e^x \) that appeared in the exercise, the derivative is remarkably straightforward because it remains \( e^x \). It makes exponential functions convenient to work with in calculus.
To differentiate effectively, you’ll often rely on some basic rules:
- Power Rule: \( \frac{d}{dx}x^n = nx^{n-1} \)
- Exponential Rule: \( \frac{d}{dx}e^x = e^x \)
- Product Rule for \( uv \): \( (uv)' = u'v + uv' \)
- Quotient Rule for \( \frac{u}{v} \): \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \)
- Chain Rule for \( f(g(x)) \): \( f'(g(x))g'(x) \)
Exponential function
The exponential function \( e^x \) is a crucial element in mathematics, especially in calculus. It's characterized by its natural behavior and constant growth rate. The base of the natural exponential function \( e \) is approximately 2.71828.
The function \( e^x \) is unique because it is its own derivative, meaning that the slope of the tangent at any point of \( e^x \) is always equal to the function value itself. This property was utilized in the exercise to simplify the indeterminate form using differentiation.
Understanding exponential functions is key to solving different mathematical problems, as they find applications in:
The function \( e^x \) is unique because it is its own derivative, meaning that the slope of the tangent at any point of \( e^x \) is always equal to the function value itself. This property was utilized in the exercise to simplify the indeterminate form using differentiation.
Understanding exponential functions is key to solving different mathematical problems, as they find applications in:
- Compound interest calculations
- Population growth models
- Radioactive decay processes
- Describing natural phenomena like temperature change
Other exercises in this chapter
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