Pregunta
What is the horizontal asymptote of \( \mathrm{f}(\mathrm{x})=e^{-\mathrm{x}}+3 \) ? \( \square \) (Type an equation:)
Ask by Park Munoz. in the United States
Jan 27,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
The horizontal asymptote is \( y = 3 \).
Solución
The horizontal asymptote of the function \( f(x) = e^{-x} + 3 \) is:
\[
y = 3
\]
**Explanation:**
As \( x \) approaches positive infinity (\( x \to \infty \)):
\[
e^{-x} \to 0
\]
Therefore:
\[
f(x) = e^{-x} + 3 \to 0 + 3 = 3
\]
Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)):
\[
e^{-x} \to \infty
\]
However, since the question typically concerns the behavior as \( x \to \infty \), the horizontal asymptote is:
\[
y = 3
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
Como
error msg
Explicar 
Simplifique esta solución Beyond the Answer
As \( x \rightarrow \infty \), the term \( e^{-x} \) approaches 0 because the exponential function decays rapidly. Therefore, \( f(x) = e^{-x} + 3 \) approaches \( 0 + 3 = 3 \). Thus, the horizontal asymptote is \( y = 3 \). As \( x \rightarrow -\infty \), the term \( e^{-x} \) grows very large since it becomes \( e^{|\infty|} \). In this case, \( f(x) \rightarrow \infty \). So for negative values of \( x \), there's no horizontal asymptote but rather the function continues to increase without bound.
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde
Hazte Premium
