Find \( \lim _{x \rightarrow 0^{+}}\left(e^{-1 / x} \sin (1 / x)-(x+2)^{3}\right) \) (if it exists) and give a careful argument showing that your answer is correct. The notation lim the uniqueness of limits. Prove that limits, if they exist, are indeed unique. That is, the suppose that \( f \) is a real valued function of a real variable, \( a \) is an accumulation point of the domain of \( f \), and \( \ell, m \in \mathbb{R} \). Prove that if \( f(x) \rightarrow \ell \) as \( x \rightarrow a \) and \( f(x) \rightarrow m \) as \( x \rightarrow a \), then \( l=m \). (Explain carefully why it was important that we require \( a \) to be an accumulation point of the domain of \( f \).) Let \( f(x)=\frac{\sin \pi x}{x+1} \) for all \( x \neq-1 \). The following information is known about a function \( g \) defined for all real numbers \( x \neq 1 \) : (i) \( g=\frac{p}{q} \) where \( p(x)=a x^{2}+b x+c \) and \( q(x)=d x+e \) for some constants \( a, b, c, d, e \); (ii) the only \( x \)-intercept of the curve \( y=g(x) \) occurs at the origin; (iii) \( g(x) \geq 0 \) on the interval \( [0,1) \) and is negative elsewhere on its domain; (iv) \( g \) has a vertical asymptote at \( x=1 \); and (v) \( g(1 / 2)=3 \). Either find lim \( g(x) f(x) \) or else show that this limit does not exist. Hints. Write an explicit formula for \( g \) by determining the constants \( a \ldots e \). Use (ii) to find \( c \); use (ii) and (iii) to find \( a \); use (iv) to find a relationship between \( d \) and \( e \); then use (v) to obtain an explicit form for \( g \). Finally look at \( f(x) g(x) \); replace sin \( \pi x \) by sin( \( (x(x-1)+\pi) \) and use the formula for the sine of the sum of two numbers.

To calculate the limit \( \lim _{x \rightarrow 0^{+}}\left(e^{-1 / x} \sin (1 / x)-(x+2)^{3}\right) \), we analyze both components as \( x \) approaches \( 0^{+} \). First, notice that \( e^{-1/x} \) converges to \( 0 \) much faster than any polynomial term grows. Therefore, \( e^{-1/x} \sin(1/x) \rightarrow 0 \). For the term \( (x+2)^3 \), we find as \( x \to 0^+ \), this approaches \( 2^3 = 8 \). Thus, the limit simplifies to: \[ \lim_{x \rightarrow 0^{+}} (0 - 8) = -8. \] Next, regarding the uniqueness of limits, we need to demonstrate that if a function \( f \) approaches two different limits \( \ell \) and \( m \) as \( x \) approaches an accumulation point \( a \), then \( \ell \) must equal \( m \). This is crucial because an accumulation point allows for \( x \) values to approach \( a \) infinitely closely from both sides, ensuring that the behavior of \( f(x) \) is consistent. If it were possible for \( f(x) \) to approach both \( \ell \) and \( m \), we would find discrepancies in the values \( f(x) \) could attain as \( x \) gets arbitrarily close to \( a \), violating the definition of a limit. For the function \( g \), we can derive its form using the given hints. Since the only x-intercept is at the origin, \( c = 0 \) leads to \( p(x) = ax^2 + bx \). The conditions suggest \( g(x) \geq 0 \) on \( [0,1) \) means \( a \) should be greater than \( 0 \). The asymptote at \( x = 1 \) gives \( e = -d \). Using \( g(1/2)=3 \), we can plug in \( x = 1/2 \) to find coefficients, eventually leading to a specific formulation for \( g(x) \). Now focusing on \( \lim_{x \to 0} g(x) f(x) \), we would need explicit equations for \( g(x) \) and analyze its behavior alongside \( f(x) \). We substitute and simplify accordingly, likely finding whether the limit exists as \( x \) goes to \( 0 \). If necessary, applying L'Hôpital's Rule may assist in resolving any indeterminate forms if they arise.