Solving $\lim_{{x}\to{0}} lnx \cdot x$: Overview & Proofs

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In summary, a limit is a mathematical concept denoted by "lim" that determines the value a function approaches as its input gets closer to a specific point. To solve limits algebraically, techniques such as factoring and using properties of limits can be used. The limit of a function can exist even if the function is undefined at that point. To evaluate the limit of a logarithmic function, logarithmic properties and algebraic techniques can be used, while also considering any restrictions or special cases. There are three main types of limits: one-sided limits, infinite limits, and limits at infinity, which can also be combined.
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tmt1
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I have

$\lim_{{x}\to{0}} lnx \cdot x$

$x$ approaches 0, and $lnx$ approaches $\infty$.

How can I reason about this.

I suppose $x$ approaches 0 more quickly than $lnx$ approaches $\infty$ , therefore it is zero. Is this accurate? How can I prove this.
 
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  • #2
I would write it as:

\(\displaystyle L=\lim_{x\to0}\left(\frac{\ln(x)}{\dfrac{1}{x}}\right)\)
 

1. What is the definition of a limit?

A limit is a mathematical concept that determines the value that a function approaches as its input approaches a certain value or point. It is denoted by the notation "lim" and is used to represent the behavior of a function near a specific input value.

2. How do you solve a limit algebraically?

To solve a limit algebraically, you can use various techniques such as factoring, simplifying, and manipulating algebraic expressions. You can also use properties of limits, such as the sum, difference, product, and quotient properties, to evaluate the limit of a function.

3. Can the limit of a function exist even if the function itself is undefined at that point?

Yes, the limit of a function can exist even if the function is undefined at that point. This is because a limit is concerned with the behavior of a function near a specific point, not necessarily at that point.

4. How do you evaluate the limit of a logarithmic function?

To evaluate the limit of a logarithmic function, you can use the logarithmic properties to rewrite the function in a simpler form. Then, you can use algebraic techniques or L'Hopital's rule to evaluate the limit. It is also important to consider any restrictions or special cases that may affect the limit.

5. What are the different types of limits?

There are three main types of limits: one-sided limits, where the input approaches the specific point from either the left or the right side; infinite limits, where the input approaches positive or negative infinity; and limits at infinity, where the input approaches infinity or negative infinity. These types of limits can also be combined, such as a one-sided limit at infinity.

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