Limits and Continuity Thu Mai, Michelle Wong, Tam Vu
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- 1. Limits and ContinuityThu Mai, Michelle Wong, Tam Vu
- 2. What are Limits?Limits are built upon the concept of infinitesimal.Instead of evaluating a function at a certain x-value,limits ask the question, What value does a functionapproaches as its input and a constant becomesinfinitesimally small? Notice how this question doesnot depend upon what f(c) actually is. The notationsfor writing a limit as x approaches a constant of thefunction f(x) is:Where c is the constant and L (if it is defined) is thevalue that the function approaches.
- 3. Evaluating Limits: Direct SubstitutionSometimes, the limit as x approaches c of f(x)is equal to f(c). If this is the case, just directlysubstitute in c for x in the limit expression, asshown below.
- 4. Dividing Out Technique1. Always start by seeing if the substitution method works.2. If, when you do so, the new expression obtained is an indeterminate form such as 0/0 try the dividing out technique!3. Because both the numerator an denominator are 0, you know they share a similar factor.4. Factor whatever you can in the given function.5. If there is a matching factor in the numerator and denominator, you can cross thru them since they one out.6. With your new, simplified function attempt the substitution method again. Plug whatever value x is approaching in for x.7. The answer you arrive at is the limit.*Note: You may need to algebraically manipulate the function.
- 5. Rationalizing Sometimes, you will come across limits with radicals in fractions.Steps1. Use direct substitution by plugging in zero for x.2. If you arrive at an undefined answer (0 in the denominator) see if there are any obvious factors you could divide out.3. If there are none, you can try to rationalize either the numerator or the denominator by multiplying the expression with a special form of 1.4. Simplify the expression. Then evaluate the rewritten limit.Ex:
- 6. Squeeze TheoremThe Squeeze Theorem states that if h(x) f(x) g(x), andthen
- 7. Special Trig Limits (memorize these) h is angle in radians area of blue: cos(h)sin(h)/2 area of pink: h/2 area of yellow: tan(h)/2Sinceby the Squeeze Theorem we can say that
- 8. Special Trig Limits Continued
- 9. Continuity and DiscontinuityA function is continuous in the interval [a,b] ifthere does not exist a c in the interval [a,b]such that:1) f(c) is undefined, or2) , or 3) The following functions are discontinuous b/c they do not fulfill ALL the properties of continuity as defined above.
- 10. Removable vs Non-removable Discontinuities A removable discontinuity exists at c if f can be made continuous byredefining f(c). If there is a removable discontinuity at c, the limit as xc exists;likewise if there is a non-removable discontinuity at c, the limit as xcdoes not exist. For this function, there is a removable discontinuity at x=3; f(3) = 4 can simply be redefined as f(3) = 2 to make the function continuous. The limit as x3 exists. For this function, there is a non-removable discontinuity at x=3; even if f(3) is redefined, the function will never be continuous. The limit as x3 does not exist.
- 11. Intermediate Value TheoremThe Intermediate Value Theorem states that iff(x) is continuous in the closed interval [a,b]and f(a) M f(b), then at least one c exists inthe interval [a,b] such that: f(c) = M
- 12. When do limits not exist?Ifthen
- 13. Vertical Asymptotesf(x) and g(x) are continuous on an open intervalcontaining c. if f(c) is not equal to 0 and g(c)= 0and theres an open interval with c which g(x) isnot 0 for all values of x that are not c, then..There is an asymptote at x = cfor
- 14. Properties of Limits Let b and c be real numbers, n be a positive integer, f and g be functions with the following limits.Sum or Difference QuotientScalar Multiple PowerProduct
- 15. Limits SubstitutionWith limits substitution (informally named soby yours truly), ifthenThis is useful for evaluating limits such as:
- 16. How Do Limits Relate to Derivatives?What is a derivative? The derivative of a function is defined as that functions INSTANT rate of change.Applying Prior Knowledge: As learned in pre-algebra, the rate of change of a function is defined by: y xApply Knowledge of Limits: Consider that a limit describes the behavior of a function as x gets closer andcloser to a point on a function from both left and right. y describes a functions rate of change. To find the functions INSTANT rate of x change, we can use limits. We can take: lim y x 0 x WHY? As the change in x gets closer and closer to 0, we can moreaccurately predict the functions INSTANT rate of change, and thus the functionsderivative.
- 17. How Do Limits Relate to Derivatives? y y y Consider that x can be rewritten as 2x 1 . (x+ x, f(x+ x)) Analyze the graph. Notice that the change (x, f(x)) in y between any two points on a function is f(x+ x) f(x). Thus: y = y2 y1 = f(x+ x) f(x) x x xSo lim y can be rewritten as lim f(x+ x) f(x) . x 0 x x0 x Therefore, the derivative of f(x) at x is given by: lim f(x+ x) f(x) x 0 x
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