Find the limits in Exercises 53–58. Write ∞ or −∞ where approp...

Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

Accepted Answer

Welcome back, everyone. Find the limit as x approaches 0 from the right of X2 minus X minus 6 divided by X cubed minus 3 x 2. A says -6, B 0 C negative infinity, and D positive infinity. Let's rewrite the limit. Lim SX approaches 0 from the right of X2 minus X minus 6. Divided by X cubed minus 3 x 2. We're going to begin with the direct substitution, which leads to 02 minus 0 minus 6 divided by 0 cubed minus 3 multiplied by 0 squared. What we notice is that we have a ratio of 6 divided by 0. Whenever we are approaching a point from one side, in other words, if we have a one-sided limit, and if we end up with a number that is non-zero divided by 0, this gives us infinity by definition, but our goal is to determine the sign of infinity positive or negative, right? So to identify the sign properly, we are going to perform a complete factorization of the function. Let's begin with the numerator. We have a quadratic polynomial, which can be factored out as X minus 3. Multiplied by X + 2, and for the denominator, we can take out X2, which leaves us with X minus 3. Now, we can cancel out the common factor which is X minus 3. And this gives us a limit as X approaches 0 from the right of X + 2 divided by X2. From here We are going to repeat the same steps with the direct substitution. X approaches 0 from the right, so we get 0 + 2, which gives us 2 in the numerator, and for the denominator, we have X2. So that's 0 from the right squared. So now we have a positive number in the numerator divided by a positive number in the denominator. Basically we're squaring a positive number which gives us a positive number. The ratio of two positive numbers is positive, meaning our infinity is positive in the initial limit. So we have determined. That the answer to this problem corresponds to the answer to the positive infinity. Thank you for watching.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

Key Concepts

Limits

Rational Functions

One-Sided Limits

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