Day 4.

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Presentation transcript:

Day 4

WARMUP—Please work the following upon arriving to class WARMUP—Please work the following upon arriving to class. THANKS for getting right to work. 

ANSWERS

ANSWERS

HW Questions:

FROM earlier this week: KNOW THESE!! What is different about these two?

Add this to your list of limits to KNOW.

A Sine Summary

A little trig practice

Answers--A little trig practice

Day 4 Infinite Limits Asymptotes AP Calculus AB Day 4 Infinite Limits Asymptotes

An Infinite Limit- Limit FAILS TO EXIST and the function increases without bound. Limit FAILS TO EXIST and the function decreases without bound.

Reemphasizing the differences . . .

Determining positive or negative infinity without a calculator . . . . . Test #s on either side of x-value and see which direction the y-values go. Pick #s that are close to the x-value.

Pick x values that are approaching 0 from the left. Pick x values that are approaching 0 from the right. -1 -¾ -½ -¼ 1 ¾ ½ ¼

Based on our work, determine the following limit.

You Try . . .

Based on our work, determine the following limit.

Calculus Definition: Vertical Asymptote If then x=a is a vertical asymptote.

Example Is there a vertical asymptote at x = 3? Justify your answer (with a CALCULUS justification).

Answer There is a vertical asymptote at x=3 because

Tying Things Together . . . Determine the following limit by showing the right hand and left hand limits using x-values. Justify, using the calculus definition, that x=3 is a vertical asymptote.

End Behavior Models What happens as x approaches ∞ or -∞? Two Options: Horizontal asymptotes Increasing or decreasing without bound

Definition: Horizontal Asymptote

End Behavior Models Comparing growth of numerator to the growth of denominator. The idea behind BOSTON. 1) The top is constant. The bottom is increasing without bound. The overall value is going to 0 Therefore, the limit is 0 2) The leading coefficients are the only factors in the slight difference in growth between the top and the bottom. So, the limit is the ratio of these coefficients. 3) The top is increasing without bound The bottom is increasing at a slower rate. Therefore, the limit is increasing without bound 4) Same as the previous except approaching negative infinity Therefore, the limit is decreasing without bound

Think of infinity as a race Consider BOSTON does not work!!!!  But, comparing growth does. The top is increasing w/out bound, making the value larger. The bottom is increasing w/out bound, making the value smaller. Which is growing faster? Who wins the race? The top or the bottom? ex grows faster than x, therefore, the limit is infinite.

Negative Exponents The top is now constant. Consider The top is now constant. Sine is bounded but ex is increasing without bound making the value smaller. Therefore, the limit is zero.

A CALCULUS justification. . . . NO BOSTON Ex: What is the horizontal asymptote of ? Answer: The horizontal asymptote is y = 0 because

Practice: