Find the Asymptotes f(x)=xe^(-x^2)
The given problem is asking to determine the asymptotes of the function f(x) = xe^(-x^2). Asymptotes are lines that the graph of a function approaches as x goes to infinity or negative infinity. There are three kinds of asymptotes: vertical, horizontal, and oblique (or slant). The question likely involves analyzing the function's behavior as x grows large in both the positive and negative directions to determine if there are any horizontal or slant asymptotes, and also examining the function for any values of x that may cause it to become undefined, which could indicate the presence of vertical asymptotes.
$f \left(\right. x \left.\right) = x e^{- x^{2}}$
Answer
Solution:
Step 1:
Determine the undefined points of $x e^{-x^2}$. The function is defined for all real numbers, so there are no points where it is undefined.
Step 2:
Identify any vertical asymptotes, which are typically found where the function is not continuous. There are no vertical asymptotes for this function.
Step 3:
To find horizontal asymptotes, calculate $\lim_{x \to \infty} x e^{-x^2}$.
Step 3.1:
Express $x e^{-x^2}$ as $\frac{x}{e^{x^2}}$ and consider $\lim_{x \to \infty} \frac{x}{e^{x^2}}$.
Step 3.2:
Invoke L'Hospital's Rule due to the indeterminate form.
Step 3.2.1:
Determine the limits of both the numerator and the denominator separately.
Step 3.2.1.1:
Evaluate $\frac{\lim_{x \to \infty} x}{\lim_{x \to \infty} e^{x^2}}$.
Step 3.2.1.2:
The limit of a polynomial with a positive leading coefficient as $x$ approaches infinity is infinity, hence $\frac{\infty}{\lim_{x \to \infty} e^{x^2}}$.
Step 3.2.1.3:
As $x^2$ goes to infinity, so does $e^{x^2}$, leading to $\frac{\infty}{\infty}$.
Step 3.2.1.4:
The expression $\frac{\infty}{\infty}$ is undefined.
Step 3.2.2:
Apply L'Hospital's Rule to the indeterminate form $\frac{\infty}{\infty}$, which allows us to take the limit of the derivatives instead.
Step 3.2.3:
Compute the derivatives of the numerator and denominator.
Step 3.2.3.1:
Take the derivatives of $x$ and $e^{x^2}$.
Step 3.2.3.2:
Apply the Power Rule to the numerator, resulting in $\lim_{x \to \infty} \frac{1}{\frac{d}{dx} e^{x^2}}$.
Step 3.2.3.3:
Use the chain rule for the derivative of the denominator.
Step 3.2.3.3.1:
Set $u = x^2$ and differentiate $e^u$ with respect to $u$ and $x^2$ with respect to $x$.
Step 3.2.3.3.2:
Differentiate $e^u$ using the Exponential Rule, resulting in $e^u \ln(e)$.
Step 3.2.3.3.3:
Substitute back $u = x^2$.
Step 3.2.3.4:
Differentiate $x^2$ using the Power Rule, giving $2x$.
Step 3.2.3.5:
Simplify the expression.
Step 3.2.3.5.1:
Rearrange the factors to obtain $\lim_{x \to \infty} \frac{1}{2 e^{x^2} x}$.
Step 3.2.3.5.2:
Simplify further to get $\lim_{x \to \infty} \frac{1}{2 x e^{x^2}}$.
Step 3.3:
Extract the constant $\frac{1}{2}$ from the limit expression.
Step 3.4:
Evaluate the limit, noting that as $x$ approaches infinity, the denominator grows without bound, making the limit $0$.
Step 3.5:
Multiply $\frac{1}{2}$ by the limit result, which is $0$.
Step 4:
The horizontal asymptote is at $y = 0$.
Step 5:
There is no slant (oblique) asymptote since the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Summarize the asymptotes found: no vertical or oblique asymptotes, and a horizontal asymptote at $y = 0$.
Knowledge Notes:
Domain of a Function: The set of all possible input values (usually 'x') for which the function is defined.
Vertical Asymptotes: Lines that the graph of a function approaches but never touches as the input values approach a certain point. They occur at points of infinite discontinuity.
Horizontal Asymptotes: Lines that the graph of a function approaches as the input values go to positive or negative infinity. They are found by evaluating the limit of the function as 'x' approaches infinity or negative infinity.
L'Hospital's Rule: A method for evaluating limits of indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ by taking the limit of the ratio of their derivatives.
Indeterminate Forms: Expressions that do not have a well-defined limit as they stand but can often be manipulated to find a limit.
Power Rule for Differentiation: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Chain Rule for Differentiation: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.
Exponential Rule for Differentiation: If $f(x) = a^x$, then $f'(x) = a^x \ln(a)$, where 'a' is a constant and 'ln' denotes the natural logarithm.
Oblique (Slant) Asymptotes: Lines that the graph of a function approaches as 'x' goes to infinity or negative infinity, and they occur when the degree of the numerator is exactly one more than the degree of the denominator.
