Limits at Infinity Explore the End Behavior of a function.

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Presentation transcript:

Limits at Infinity Explore the End Behavior of a function

End Behavior The following is very important.

End Behavior The following is very important. When the limit at infinity exists, it is known as the horizontal asymptote.

Example 2

Example 3 The following limits are one definition of e.

Polynomials We can generalize the end behavior of polynomials.

Polynomials We can generalize the end behavior of polynomials. If the leading coefficient of the polynomial is positive:

Polynomials We can generalize the end behavior of polynomials. If the leading coefficient of the polynomial is negative:

Polynomials The end behavior of a polynomial matches the end behavior of its highest degree term.

Polynomials The end behavior of a polynomial matches the end behavior of its highest degree term. For example:

Limits of Rational Functions We will have three different situations.

Limits of Rational Functions We will have three different situations. 1.The degree of the numerator and the denominator is the same.

Limits of Rational Functions We will have three different situations. 1.The degree of the numerator and the denominator is the same. 2.The degree of the denominator is higher than the degree of the numerator.

Limits of Rational Functions We will have three different situations. 1.The degree of the numerator and the denominator is the same. 2.The degree of the denominator is higher than the degree of the numerator. 3.The degree of the numerator is higher than the degree of the denominator.

Same Degree Find:

Same Degree Find: We will divide the numerator and denominator by the highest power of x in the denominator.

Same Degree Find: We will divide the numerator and denominator by the highest power of x in the denominator.

Same Degree When the numerator and the denominator are the same degree, your answer is a number, and it is the quotient of the leading coefficients.

Denominator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Denominator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Denominator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Denominator is higher power Conclusion: When the denominator has the highest power of x, the answer is zero. For fractions, if the denominator accelerates faster than the numerator, the limit is zero.

Numerator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Numerator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Numerator is higher power We start the same way; divide the numerator and denominator by the highest power of x in the denominator. Find:

Numerator is higher power Conclusion: When the numerator is the higher power of x, the answer is. In the past example, the function reacts like a negative second degree equation, so the answer is

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find: If x is approaching positive infinity, the absolute value of x is just x.

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find: If x is approaching negative infinity, the absolute value of x is the opposite of x.

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits involving radicals Find:

Limits of exponential and logarithmic functions

Let’s look at e x.

Limits of exponential and logarithmic functions Let’s look at e -x

Homework Section 1.3 Pages odd