Limit x mendekati ketakhinggan (4x²-5x+3-2x+1)

To find the limit of the function \( 4x^2 - 5x + 3 - 2x + 1 \) as \( x \) approaches infinity (\( x \to \infty \)), let's first simplify the expression:

\[
4x^2 - 5x + 3 - 2x + 1 = 4x^2 - 7x + 4.
\]

Now, to find the limit as \( x \) approaches infinity, we focus on the term with the highest power of \( x \), which is \( 4x^2 \). As \( x \) becomes very large, the other terms (\( -7x \) and \( +4 \)) become insignificant in comparison to \( 4x^2 \). Therefore, the expression \( 4x^2 - 7x + 4 \) behaves similarly to \( 4x^2 \) as \( x \to \infty \).

Mathematically, we can express this limit as:

\[
\lim_{x \to \infty} (4x^2 - 7x + 4) = \lim_{x \to \infty} 4x^2.
\]

Since the dominant term is \( 4x^2 \), the limit is:

\[
\lim_{x \to \infty} 4x^2 = \infty.
\]

Therefore, 

\[
\lim_{x \to \infty} (4x^2 - 5x + 3 - 2x + 1) = \infty.
\]

This means the function increases without bound as \( x \) approaches infinity.

Would you like a detailed step-by-step explanation, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would you find the limit of \( \frac{4x^2 - 7x + 4}{x^2} \) as \( x \) approaches infinity?
2. What happens to the limit if the leading coefficient was negative, for example, \( -4x^2 \)?
3. How does the degree of the polynomial affect the limit as \( x \) approaches infinity?
4. How do you find the limit of a rational function as \( x \) approaches infinity?
5. What is the limit of \( 4x^2 - 7x + 4 \) as \( x \) approaches negative infinity?

**Tip:** When evaluating limits as \( x \) approaches infinity, always look for the term with the highest power of \( x \), as it will dominate the behavior of the function.

Limit x mendekati ketakhinggan (4x² - 5x + 3 - 2x + 1)

Explore the detailed solution to finding the limit of the polynomial expression 4x² - 5x + 3 - 2x + 1 as x approaches infinity. This solution simplifies the equation and explains how the dominant term affects the limit. Suitable for advanced high school students or early college learners studying calculus.

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