# Solving Limits: Help Me Find a Limit!

• stefaneli
In summary, the conversation is about finding a limit involving the imaginary unit j and the expression \frac{\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)}{(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta))^2+ \sqrt2(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)))+1}. The solution is ∞ exp( \frac{∏}{4} - \theta) as ρ→0, which is derived by expanding the denominator and simplifying the expression.
stefaneli

## Homework Statement

I don't know how to find a limit, and it's bothering me for a few hours now.
Can someone help me?
j - imaginary unit

## Homework Equations

$\lim_{\rho \to 0}{\frac{\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)}{(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta))^2+ \sqrt2(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)))+1}}$

## The Attempt at a Solution

Solution is:
$∞ exp( \frac{∏}{4} - \theta)$

stefaneli said:

## Homework Statement

I don't know how to find a limit, and it's bothering me for a few hours now.
Can someone help me?
j - imaginary unit

## Homework Equations

$\lim_{\rho \to 0}{\frac{\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)}{(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta))^2+ \sqrt2(\frac{\sqrt{2}}{2}(-1+j)+\rho \exp(j\theta)))+1}}$

## The Attempt at a Solution

Solution is:
$∞ exp( \frac{∏}{4} - \theta)$

The denominator approaches 0 and the numerator doesn't. It doesn't have a limit.

To be exact...
$\rho \rightarrow 0+$

The solution I've written is correct for sure.:)

stefaneli said:
To be exact...
$\rho \rightarrow 0+$

The solution I've written is correct for sure.:)

Ok, let's write $a=\frac{\sqrt{2}}{2} (-1+j)$ and $r=\rho exp( j \theta)$ then your expression is $$\frac{a+r}{(a+r)^2+\sqrt{2} (a+r)+1}$$
If you expand the denominator, and putting in the value for a, you get $$\frac{a+r}{r^2+j \sqrt{2} r}$$
As ρ→0 you can ignore the r in the numerator and the r^2 in the denominator. Now you just have to express $$\frac{a}{j \sqrt{2} r}$$ as a magnitude and phase. Can you take it from there? It's not really a limit, it's a limiting behavior.

Thanks...it helped me:)

## 1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It represents the value that a function "approaches" as its input gets closer and closer to a specific value, but may not necessarily equal at that specific value.

## 2. How do I solve a limit?

To solve a limit, you can use algebraic techniques such as factoring and simplifying, as well as various limit laws and theorems. You can also use graphical and numerical methods, such as using a graphing calculator or creating a table of values. In some cases, you may need to use more advanced techniques such as L'Hopital's rule or the squeeze theorem.

## 3. What are the common types of limits?

The common types of limits include polynomial limits, rational limits, trigonometric limits, exponential limits, and logarithmic limits. Each type of limit may require different techniques to solve, but the fundamental concept remains the same.

## 4. How do I know if a limit does not exist?

A limit does not exist if the function has a discontinuity, such as a hole or vertical asymptote, at the point where the limit is being evaluated. It can also not exist if the right-hand limit and left-hand limit approach different values, or if the function oscillates or becomes unbounded as the input approaches the specific value.

## 5. Why are limits important?

Limits are important because they allow us to understand the behavior of a function at a particular point, even if the function is not defined at that point. They are essential in calculus, as they are used to define derivatives and integrals, and play a crucial role in understanding the fundamental concepts of continuity and differentiability.

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