AI Data Science - 1st Semester / 20024 - PUCSP University - Math Repository - Professor Eric Bacconi Gonçalves
Simplified Form: The numerator
Substituting ( x ) with 3, we get:
Explanation: The limit as ( x ) approaches 3 for the function
This is because the factor ( x - 3 ) in the denominator cancels out with the same factor in the numerator, leaving ( x + 3 ) which evaluates to 6 when ( x ) is 3.
Simplified Form: The numerator
This allows us to simplify the expression by canceling out the common factor of:
When we substitute ( x ) with -7, the expression simplifies to: 7 - (-7) = 14
Explanation: The limit of the function
This result is obtained because after canceling the common factor, we are left with ( 7 - x ), which equals 14 when ( x ) is -7.
To solve the limit, we can factor the numerator:
So the limit becomes:
We can cancel out the ((x - 1)) terms:
Now, we can directly substitute ( x = 1 ):
Therefore, the limit is:
To calculate the limit, we can simplify the expression by factoring the numerator, which is a perfect square trinomial. Factoring (x^2 - 2x + 1), we get ((x - 1)(x - 1)). The denominator is already in factored form as (x - 1). Thus, the function simplifies to:
After canceling out the common term ( x - 1 ), we are left with:
Since there are no more terms that depend on ( x ), this simplifies to:
The limit of the function as ( x ) approaches 1 is simply 0.
To solve this limit, we need to factor the denominator and simplify the expression. The denominator ( x^2 - 4 ) can be factored into ( (x + 2)(x - 2) ), which allows us to cancel out the ( x - 2 ) term in the numerator:
Substituting ( x = 2 ) into the simplified expression, the final value:
The limit of the function as ( x ) approaches 1 is simply
This limit can be solved using factorization and polynomial division:
The limit of the function as ( x ) approaches 1 is simply
In this case, we can use L'Hôpital's rule, as the limit is of the form (\frac{0}{0}) or (\frac{\infty}{\infty}) when (x) tends to infinity.
The limit of the expression is
The limit as ( x ) approaches infinity for (
is: $\lim_{{x \to \infty}} \frac{1}{{x^2}} = 0$
As ( x ) increases without bound, the value of ( \frac{1}{{x^2}} ) approaches 0 because the denominator grows much faster than the numerator.
(
The limit as ( x ) approaches negative infinity for (
As ( x ) decreases without bound, the value of (
The limit as ( x ) approaches infinity for ( x^4 ) is: grows at an increasing rate and approaches infinity for ( x^4 ) is:
Similar to the previous expressions, the term ( 2x^5 ) grows at a faster rate than the others, causing the expression to approach infinity.
The limit as ( x ) approaches negative infinity for ( 2x^4 - 3x^3 + x + 6 ) is:
Even though ( x ) is negative, the highest power term ( x^4 ) will still lead the expression to increase without bound because the even power makes it positive.
2x^5 - 3x^2 + 6
The limit as ( x ) approaches infinity for ( 2x^5 - 3x^2 + 6 ) is:
The limit as ( x ) approaches negative infinity for ( 2x^4 - 3x^3 + x + 6 ) is:
Even though ( x ) is negative, y.
The given function is a polynomial function of the form:
As x approaches infinity, the highest power of x in the function dominates the value of the function. This means that we can ignore all the lower-order terms, and simply consider the behavior of the highest-order term. In this case, the highest-order term is 2x4. As x approaches infinity, x4 also approaches infinity, and so the function f(x) also approaches infinity.
Therefore, the limit of the function as x approaches infinity is infinity. We can write this mathematically as:
The given function is a rational function of the form
, where n > m. As x approaches infinity, the highest power of x in the numerator dominates the value of the numerator, and the highest power of x in the denominator dominates the value of the denominator. This means that we can ignore all the lower-order terms, and simply consider the behavior of the highest-order terms.
In this case, the highest-order term in the numerator is 2x4, and the highest-order term in the denominator is x3.
As x approaches infinity, 2x4 grows much faster than x3, and so the function f(x) approaches zero.