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.

${((1 + x))}^{\frac{1}{x}}$

Answer

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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.