13 This question requires the language of limits from the first two pages of this section. a In Question 4 above, the line \( y=0 \) is an asymptote to \( y=3^{x} \). Write a statement using limits to justify this. b In Question 4 b, the line \( y=0 \) is an asymptote to the exponential curve \( y=2^{-x} \). Write a statement using limits to justify this. c In Question \( \mathbf{1} \), the lines \( y=0 \) and \( x=0 \) are asymptotes to the hyperbola \( y=\frac{2}{x} \). Write four statements using limits to justify this.

As \( x \) approaches infinity in the function \( y = 3^x \), the value of \( y \) grows without bound, which we can express as \( \lim_{x \to \infty} 3^x = \infty \). Conversely, as \( x \) approaches negative infinity, \( y \) approaches zero, so we write \( \lim_{x \to -\infty} 3^x = 0 \). This indicates that the line \( y = 0 \) is a horizontal asymptote. For the function \( y = 2^{-x} \), a similar approach applies. As \( x \to \infty \), the exponential function \( 2^{-x} \) tends towards zero, expressed as \( \lim_{x \to \infty} 2^{-x} = 0 \). On the other hand, as \( x \to -\infty \), \( 2^{-x} \) increases towards infinity, \( \lim_{x \to -\infty} 2^{-x} = \infty \). Thus, the line \( y = 0 \) serves as a horizontal asymptote here as well. Now, for the hyperbola \( y = \frac{2}{x} \), as \( x \) approaches infinity, the expression converges to zero, so we state \( \lim_{x \to \infty} \frac{2}{x} = 0 \). Conversely, as \( x \) approaches negative infinity, we find \( \lim_{x \to -\infty} \frac{2}{x} = 0 \). Additionally, as \( x \) approaches zero from the right, the function heads towards infinity: \( \lim_{x \to 0^+} \frac{2}{x} = \infty \), and as \( x \) approaches zero from the left, it also approaches negative infinity, \( \lim_{x \to 0^-} \frac{2}{x} = -\infty \). Hence, both lines \( y = 0 \) and \( x = 0 \) act as asymptotes to the hyperbola.