Limit of a Geometric Sequence: How to Evaluate?

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In summary, a simple limit is a mathematical concept denoted by "lim" that represents the value a function approaches as its input approaches a specific value. To evaluate a simple limit, techniques such as direct substitution, factoring, and rationalization can be used along with an understanding of limit laws and the Squeeze Theorem. There are common types of simple limits, including polynomial, rational function, exponential function, and trigonometric function limits, each with its own set of rules for evaluation. Evaluating simple limits is important in understanding a function's behavior and solving problems in various fields. L'Hopital's rule can only be used in certain situations and cannot be applied to all simple limits.
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I have this limit:

$$\lim_{{n}\to{\infty}} {(\frac{2}{3})}^{n}$$

I know the answer is 0 but how can I evaluate this?
 
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Can you agree that $\lim_{n\to\infty}\dfrac{1}{e^n}=0$, without calculation?

If so,

$$\lim_{n\to\infty}\left(\dfrac23\right)^n=\lim_{n\to\infty}e^{n\ln(2/3)}=\lim_{n\to\infty}e^{-n\ln(3/2)}$$

$$=\lim_{n\to\infty}\left(\dfrac{1}{e^n}\right)^{\ln(3/2)}=0^{\ln(3/2)}=0$$
 

What is a simple limit?

A simple limit is a mathematical concept that represents the value that a function approaches as its input approaches a specific value. It is denoted by the symbol "lim" and is commonly used in the study of calculus.

How do you evaluate a simple limit?

To evaluate a simple limit, you can use several techniques such as direct substitution, factoring, and rationalization. It is also important to understand the properties of limits, such as the limit laws and the Squeeze Theorem, to correctly evaluate a limit.

What are the common types of simple limits?

The most common types of simple limits include polynomial limits, rational function limits, exponential function limits, and trigonometric function limits. Each type has its own set of rules and techniques for evaluation.

Why is it important to evaluate simple limits?

Evaluating simple limits is crucial in understanding the behavior of a function and its graph. It also allows us to solve various problems in mathematics, physics, and engineering that involve finding the maximum or minimum values of a function.

Can you use L'Hopital's rule to evaluate all simple limits?

No, L'Hopital's rule can only be used to evaluate simple limits in certain situations, such as when the limit results in an indeterminate form (e.g. 0/0 or ∞/∞). It cannot be used for evaluating limits involving trigonometric functions or limits at infinity.

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