Q. 2 - Calculus

Question

Simple limit calculations: Determine each of the limits that follow. You should be able to solve all of these very quickly by thinking about the graphs of the functions.

$$\begin{gathered} \underset{x\to \infty }{lim}{2}^x \\ \underset{x\to \infty }{lim}{x}^{-5} \\ \underset{x\to -\infty }{lim}{e}^{3x} \\ \underset{x\to 0}{lim}\frac{1}{x^2} \\ \underset{x\to 0^{+}}{lim}\ln x \\ \underset{x\to \infty }{lim}{\left(\frac{1}{2}\right)}^x \\ \underset{x\to \frac{\mathrm{\pi }}{2}}{lim}\tan x \\ \underset{x\to \infty }{lim}\sin x \\ \underset{x\to \infty }{lim}{\log }_{\frac{1}{2}}x \end{gathered}$$

Step-by-Step Solution

Answer

The values of the given forms are:

$$\begin{gathered} \underset{x\to \infty }{lim}{2}^x=\infty \\ \underset{x\to \infty }{lim}{x}^{-5}=0 \\ \underset{x\to -\infty }{lim}{e}^{3x}=0 \\ \underset{x\to 0}{lim}\frac{1}{x^2}=Not defined \\ \underset{x\to 0^{+}}{lim}\ln x=Not defined \\ \underset{x\to \infty }{lim}{\left(\frac{1}{2}\right)}^x=0 \\ \underset{x\to \frac{\mathrm{\pi }}{2}}{lim}\tan x=Not defined \\ \underset{x\to \infty }{lim}\sin x=Not defined \\ \underset{x\to \infty }{lim}{\log }_{\frac{1}{2}}x=Not defined. \end{gathered}$$

Step 1. Given Information.

The function:

$$\begin{gathered} \underset{x\to \infty }{lim}{2}^x \\ \underset{x\to \infty }{lim}{x}^{-5} \\ \underset{x\to -\infty }{lim}{e}^{3x} \\ \underset{x\to 0}{lim}\frac{1}{x^2} \\ \underset{x\to 0^{+}}{lim}\ln x \\ \underset{x\to \infty }{lim}{\left(\frac{1}{2}\right)}^x \\ \underset{x\to \frac{\mathrm{\pi }}{2}}{lim}\tan x \\ \underset{x\to \infty }{lim}\sin x \\ \underset{x\to \infty }{lim}{\log }_{\frac{1}{2}}x \end{gathered}$$

Step 2. Graphs of the function and its limits.

The graph of the function $2^x$ is:

From the graph, $\underset{x\to \infty }{lim}{2}^x=\infty$.

The graph of the function $x^{-5}$ is:

From the graph, $\underset{x\to \infty }{lim}{x}^{-5}=0$

The graph of the function $e^{3x}$ is:

From the graph, $\underset{x\to -\infty }{lim}{e}^{3x}=0$

The graph of the function $\frac{1}{x^2}$ is:

From the graph, $\underset{x\to 0}{lim}\frac{1}{x^2}=Not defined$

Step 3. Determine the function and its limits.

The graph of the function $\ln x$ is:

From the graph, $\underset{x\to 0^{+}}{lim}\ln x=Not defined$

The graph of the function ${\left(\frac{1}{2}\right)}^x$ is:

From the graph, $\underset{x\to \infty }{lim}{\left(\frac{1}{2}\right)}^x=0$

The graph of the function $\tan x$ is:

From the graph, $\underset{x\to \frac{\mathrm{\pi }}{2}}{lim}\tan x=Not defined$

The graph of the function $\sin x$ is:

From the graph, $\underset{x\to \infty }{lim}\sin x=Not defined$

The graph of the function ${\log }_{\frac{1}{2}}x$ is;

From the graph, $\underset{x\to \infty }{lim}{\log }_{\frac{1}{2}}x=Not defined$