Q. 2s - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

1. Two limits that are initially in an indeterminate form but can be solved with the Cancellation Theorem.
2. Two limits that can be solved with the Squeeze Theorem.
3. Two limits that we do not yet know how to calculate.

Step-by-Step Solution

Answer

Part(a) The examples are,

 $$\begin{gathered} \underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4 \\ \underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(2x\right)}{\cos x}=2 \end{gathered}$$.

Part(b) The examples are,

$$\begin{gathered} \underset{n\to 0}{\lim }\left(n^2\sin \left(\frac{1}{n}\right)\right)=0 \\ \underset{n\to 0}{\lim }\left(n\cos \left(\frac{1}{n}\right)\right)=0 \end{gathered}$$

Part(c) The examples are,

$$\begin{gathered} \underset{x\to \infty }{\lim }\left(\sin \right(x\left)\right) \\ \underset{x\to \infty }{\lim }(1-\cos (x\left)\right) \end{gathered}$$

Part(a) Step 1. Given Information.

We are given statements.

Part(a) Step 2. Examples of statement.

The examples for two limits that are initially in an indeterminate form but can be solved with the Cancellation Theorem are,

$$\begin{gathered} \underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4 \\ \underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(2x\right)}{\cos x}=2 \end{gathered}$$

Part(b) Step 1. Examples for given statement.

The examples for two limits that can be solved with the Squeeze Theorem are,

$$\begin{gathered} \underset{n\to 0}{\lim }\left(n^2\sin \left(\frac{1}{n}\right)\right)=0 \\ \underset{n\to 0}{\lim }\left(n\cos \left(\frac{1}{n}\right)\right)=0 \end{gathered}$$

Part(c) Step 1. Examples for given statement.

The examples for two limits that we do not yet know how to calculate are,

$$\begin{gathered} \underset{x\to \infty }{\lim }\left(\sin \right(x\left)\right) \\ \underset{x\to \infty }{\lim }(1-\cos (x\left)\right) \end{gathered}$$