3 handouts section2-2
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1. The document defines the limit of a function as making the function values arbitrarily close to a limit L by taking the input value closer to a specific value. 2. It discusses one-sided limits and how the limit exists only when the left and right side limits are equal. Infinite limits result when direct substitution makes the denominator equal to zero. 3. Vertical asymptotes occur when the graph shoots to positive or negative infinity as the input approaches a particular value, or when factoring a rational function results in the denominator equaling zero at that input value.
3 handouts section2-2
- 1. Section 2.2 The limit of a function Learning outcomes After completing this section, you will inshaAllah be able to 1. an idea about the meaning and definition of limit 2. get an idea about the meaning of one-sided limit 3. know the meaning of existence of limit 4. understand and compute infinite limits 5. find vertical asymptotes of a function 12.2
- 2. Meaning of limit • We learn by an example Example: To understand 3 1 1 lim 1x x x→ − − • We use the graph of the function and a table of values near x=1 • By graph • By table A table of values of the function near x=1 is x 0.5 0.75 0.9 0.99 0.999 1 1.001 1.01 1.1 1.25 1.5 y 1.750 2.313 2.710 2.970 2.997 not defined 3.003 3.030 3.310 3.813 4.750 • Conclusion: As the values of x get closer and closer to 1, we see that the values of y get closer and closer to 3. • That means 3 1 1 lim 3. 1x x x→ − = − 22.2 See class explanation for more understanding (Specially “Run-and-Hit idea”) Note: The function 3 1 1 x y x − = − is not defined at x=1. In fact the circle ‘o’ in the graph indicates that this point is missing from the graph. Question What happens to the values of 3 1 1 x y x − = − as x gets closer to 1
- 3. Understanding the definition of limit 32.2 We say that lim ( ) x a f x L → = if we can make ( )f x as close to Las we like by taking x sufficiently close to a This is read as “limit of ( )f x is L, as x approaches a ” See example 1 done in class For practical purpose, it is best to use Run – and – Hit Idea explained in class
- 4. One-sided limits • We learn by an example • Consider a function ( )y f x= given by the following graph. • From the graph we have: When we approach 0 from the left side, the value of f(x) approaches –1. When we approach 0 from the right side, the value of f(x) approaches 1. • We describe this situation by saying The limit of f(x) is –1 as x approaches 0 from left and write as 0 lim ( ) 1 x f x− → = − , and The limit of f(x) is 1 as x approaches 0 from right and write as 0 lim ( ) 1 x f x+ → = 42.2 Left side limit Obtained by approaching from left Right side limit Obtained by approaching from right Notation lim ( ) x a f x− → Notation lim ( ) x a f x+ → Q. Will we always get different answers of left and right limits? Ans. No. Check what happened in example 1 above.
- 5. Meaning of existence of a limit • We see from above that some time the left side & right side limit will be same, and some time these will be different. Example: Q: Does 0 lim ( ) x f x → exist? Q. Does 1 lim ( ) x f x → exist? 52.2 • lim ( ) x a f x → exists and lim ( ) x a f x L → = if left side limit = right side limit i.e. lim ( ) x a f x−→ = lim ( ) x a f x L+→ = See examples 2, 3 done in class
- 6. What are infinite limits? (those limits whose answer is ∞or −∞) • Look at the following graphs to understand the meaning of infinite limits Computing infinite limits lim ( ) x a f x → i.e. the answer is ∞or −∞ 62.2 See class discussion & explanation This generally happens when directly substituting x a= gives 0 k⎛ ⎞ ⎜ ⎟ ⎝ ⎠ form ( )0k ≠ Trick Look at the sign and get the answer as ∞ or −∞ See examples 4, 5, 6, 7 done in class Note usefulness of factorization
- 7. Vertical Asymptotes • Look at the following graphs again. 72.2 What’s special about line x=a It runs (very close &) parallel to graph up to ±∞ What happens to graph when we get near the value x=a The graph either shoots up to ∞ or shoots down to −∞ A vertical line x a= is called vertical asymptote of graph of ( )f x if one of the following is true lim ( ) or x a f x− → = ∞ − ∞ lim ( ) or x a f x+ → = ∞ − ∞ lim ( ) or x a f x → = ∞ − ∞ Vertical asymptotes for ( ) ( ) ( ) P x f x Q x = x a= is a vertical asymptote if ( ) 0 and ( ) 0Q a P a= ≠ Special situation for rational functions See examples 8, 9 done in class But it is better to compute limits to get complete idea of the situation End of 2.2 Note usefulness of factorization