Graph (1+x)^(1/x)
The question requires you to consider the graph of the function f(x) = (1+x)^(1/x), which is an exponential function with a variable base of 1+x and a variable exponent of 1/x. You would be expected to examine how this function behaves as x varies, identifying key features such as its continuity, limits, asymptotic behavior, local extrema, and inflection points. Moreover, it may be relevant to discuss the function's behavior as x approaches 0, as well as the end behavior as x approaches positive or negative infinity. To answer this question fully, you would need to create a visual representation of the function on a set of axes, detailing these key characteristics.
$\left(\left(\right. 1 + x \left.\right)\right)^{\frac{1}{x}}$
Answer
Solution:
Step 1:
Identify the values for which the function $f(x) = (1 + x)^{\frac{1}{x}}$ is not defined. This occurs when $x = 0$.
Step 2:
Determine if there are any vertical asymptotes. There are none for this function.
Step 3:
To find the horizontal asymptote, calculate the limit of $f(x)$ as $x$ approaches infinity.
Step 3.1:
Transform the limit using logarithmic properties for easier evaluation.
Step 3.1.1:
Express the function as $e^{\ln((1 + x)^{\frac{1}{x}})}$ and take the limit as $x$ approaches infinity.
Step 3.1.2:
Simplify the natural logarithm by bringing the exponent $\frac{1}{x}$ in front of the $\ln$ function.
Step 3.2:
Proceed with the limit calculation.
Step 3.2.1:
Move the limit operation into the exponent.
Step 3.2.2:
Simplify the expression inside the limit.
Step 3.3:
Apply L'Hospital's Rule to resolve the indeterminate form $\frac{\infty}{\infty}$.
Step 3.3.1:
Separate the limit of the numerator and the denominator.
Step 3.3.1.1:
Evaluate the limits separately.
Step 3.3.1.2:
Recognize that the logarithm function approaches infinity as its argument goes to infinity.
Step 3.3.1.3:
Acknowledge that the limit of a polynomial with a positive leading coefficient also goes to infinity.
Step 3.3.1.4:
Understand that infinity over infinity is an indeterminate form.
Step 3.3.2:
Since we have an indeterminate form, apply L'Hospital's Rule.
Step 3.3.3:
Find the derivatives of the numerator and denominator.
Step 3.3.3.1:
Differentiate both the numerator and denominator.
Step 3.3.3.2:
Apply the Chain Rule for differentiation.
Step 3.3.3.2.1:
Set $u = 1 + x$ to use the Chain Rule.
Step 3.3.3.2.2:
The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$.
Step 3.3.3.2.3:
Substitute back $1 + x$ for $u$.
Step 3.3.3.3:
Apply the Sum Rule to differentiate $1 + x$.
Step 3.3.3.4:
Since the derivative of a constant is zero, the derivative of $1$ is $0$.
Step 3.3.3.5:
Combine the derivatives.
Step 3.3.3.6:
Use the Power Rule for differentiation.
Step 3.3.3.7:
Multiply the fraction by $1$.
Step 3.3.3.8:
Rearrange the terms for clarity.
Step 3.3.3.9:
Apply the Power Rule again to find the derivative of $x$.
Step 3.3.4:
Multiply the numerator by the reciprocal of the denominator.
Step 3.3.5:
Simplify the expression by multiplying.
Step 3.4:
Since the numerator approaches a real number and the denominator grows without bound, the limit is $0$.
Step 3.5:
Any number raised to the power of $0$ is $1$.
Step 4:
The horizontal asymptote is at $y = 1$.
Step 5:
There is no slant (oblique) asymptote, as the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Summarize the asymptotes of the function: no vertical asymptotes, a horizontal asymptote at $y = 1$, and no oblique asymptotes.
Knowledge Notes:
Undefined Points: A function is undefined at points where it cannot produce a valid output. For example, division by zero or taking the logarithm of a negative number.
Asymptotes: These are lines that the graph of a function approaches but never touches. Vertical asymptotes occur at points of infinite discontinuity, horizontal asymptotes occur when the function approaches a constant value as $x$ approaches infinity or negative infinity, and oblique asymptotes occur when the function approaches a line that is not horizontal.
Limits: The concept of a limit helps us understand the behavior of functions as they approach specific points or infinity.
Logarithmic Properties: These properties allow us to simplify complex expressions, such as turning exponents into multipliers for the logarithm.
L'Hospital's Rule: This rule is used to evaluate limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It states that the limit of a ratio of two functions can be found by taking the limit of the ratio of their derivatives.
Chain Rule: A rule for finding the derivative of a composite function. It states that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.
Power Rule: A basic rule of differentiation that states the derivative of $x^n$ is $n \cdot x^{n-1}$.
Sum Rule: A rule that allows us to differentiate a function that is the sum of two or more functions by differentiating each function individually and then summing the results.
Multiplication by Reciprocal: Multiplying by a reciprocal is the same as dividing by the original number. This can simplify expressions, especially when dealing with limits.
Exponential Functions: The function $e^x$ is its own derivative, and $e^0 = 1$. This is a key property used when evaluating certain limits.
