Assuming that limits as
x → ∞ have the properties for limits as x → c, use algebraic manipulations to evaluate lim x → ∞ for the function.
a. f(x) = x − 2/2 + 6x^2
b. f(x) = x^2 + 6/x + 7
Please help! I do not know how to start with this question.
I got zero for part a by plugging the function into a calculator but I still do not get part b.
Nevermind, I got it! I just needed to zoom out more on my graph.
To evaluate limits as x approaches infinity, you can use algebraic manipulations to rewrite the function in a way that allows you to apply the limit properties.
Let's start with part (a):
a. f(x) = (x - 2)/(2 + 6x^2)
To evaluate lim x → ∞ f(x), we need to look at the highest power of x in the denominator and numerator. In this case, the highest power of x in the denominator is x^2, and the highest power of x in the numerator is x.
When x approaches infinity, the terms with lower powers become insignificant compared to the terms with higher powers. So, we can ignore the terms x^2 in the denominator and x^0 (which is just 1) in the numerator.
Now, we have:
lim x → ∞ (x - 2)/(2 + 6x^2)
≈ lim x → ∞ (x/2x^2) (ignoring lower power terms)
≈ lim x → ∞ (1/2x) (canceling out common factor of x)
Now, we can take the limit:
lim x → ∞ (1/2x) = 0
So, lim x → ∞ f(x) = 0.
Now let's move on to part (b):
b. f(x) = (x^2 + 6)/(x + 7)
Again, we look at the highest power of x in the denominator and numerator. In this case, the highest power of x is x^2 in the numerator, and x in the denominator.
So, when x approaches infinity, we need to divide both the numerator and denominator by x^2:
lim x → ∞ (x^2 + 6)/(x + 7)
≈ lim x → ∞ (1 + 6/x^2)/(1/x + 7/x^2) (dividing numerator and denominator by x^2)
≈ lim x → ∞ (1 + 6/x^2)/(1/x + 7/x^2) (simplifying)
Now, we can take the limit:
lim x → ∞ (1 + 6/x^2)/(1/x + 7/x^2)
= (1 + 0)/(0 + 0) (taking the limit as x approaches infinity)
= 1/0
Since we have a denominator of 0, the limit is undefined.
So, lim x → ∞ f(x) is undefined for the function in part (b).
I hope this helps you understand how to evaluate limits as x approaches infinity using algebraic manipulations! Let me know if you have any further questions.