1. The problem is to understand the notation and meaning of limits at infinity for a function $\phi(x)$. 2. When we write $$\lim_{x \to -\infty} \phi(x) = M,$$ it means that as $x$ decreases without bound (goes to negative infinity), the values of $\phi(x)$ get arbitrarily close to the number $M$. 3. Similarly, when we write $$\lim_{x \to +\infty} \phi(x) = M,$$ it means that as $x$ increases without bound (goes to positive infinity), the values of $\phi(x)$ get arbitrarily close to the number $M$. 4. This concept is important in calculus and analysis to describe the end behavior of functions. 5. The notation uses the limit symbol $\lim$ with the variable $x$ approaching $-\infty$ or $+\infty$ to indicate the direction of approach. 6. The number $M$ is called the limit or the horizontal asymptote if the function approaches it as $x$ goes to infinity or negative infinity. 7. Understanding these limits helps in graphing functions and analyzing their long-term behavior.