1. Let's start by stating the problem: We want to find the limits of some exponential, logarithmic, and trigonometric functions as the variable approaches a certain value. 2. For exponential functions, the general form is $f(x) = a^x$ where $a > 0$ and $a \neq 1$. Important rule: as $x \to \infty$, if $a > 1$, then $a^x \to \infty$; if $0 < a < 1$, then $a^x \to 0$. 3. For logarithmic functions, the general form is $f(x) = \log_a(x)$ where $a > 0$ and $a \neq 1$. Important rule: as $x \to 0^+$, $\log_a(x) \to -\infty$; as $x \to \infty$, $\log_a(x) \to \infty$. 4. For trigonometric functions like $\sin x$ and $\cos x$, limits depend on the point approached. For example, $\lim_{x \to 0} \sin x = 0$ and $\lim_{x \to 0} \frac{\sin x}{x} = 1$. 5. Let's evaluate an example limit for each type: - Exponential: $\lim_{x \to \infty} 2^x = \infty$ because base 2 is greater than 1. - Logarithmic: $\lim_{x \to 0^+} \log_2 x = -\infty$ because logarithm approaches negative infinity near zero. - Trigonometric: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ by the standard limit rule. 6. These limits help us understand the behavior of these functions near critical points. Final answers: - $\lim_{x \to \infty} 2^x = \infty$ - $\lim_{x \to 0^+} \log_2 x = -\infty$ - $\lim_{x \to 0} \frac{\sin x}{x} = 1$