- Continuity 1 (19.64 KiB) Viewed 33 times
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A function f(x) is said to have a jump discontinuity at x=a if: 1. x→a−limf(x) exists. 2. x→a+limf(x) exists. 3. The left and right limits are not equal. Let f(x)={6x−6,x+83, if x<6 if x≥6 Show that f(x) has a jump discontinuity at x=6 by calculating the limits from the left and right at x=6. x→6−limf(x)= z→5+limf(x)= Now, for fun, try to graph f(x).
Find the value of the constant m that makes the following function continuous on (−∞,∞). f(x)={mx−14x2+2x−7 if x<−7 if x≥−7 m= Now draw a graph of f.
Shown below are six statements about functions. Match each statement to one of the functions shown below which BEST matches that statement. 1. z→2limf(x) and x→9−limf(x) both exist and are fnite, but they are not equal. 2. The graph of y=f(x) has vertical tangent line at (9,f(9)) 3. z→0lim⋅f(x)=−∞. 4. 2→+9limf(x) exists but t→1limf(x) does not. 5. c+∞limf(x)=∞. 6. x→0limf(x) exists but f is not continuous at 9 . A. f(x)=4x−9 B. f(x)=(x−θ)1 C. f(x)=⎩⎨⎧cos(x−61)02x+36 If x<9 if x=9 if x>9 D. f(x)=⎩⎨⎧2x036−2z if x<9 if x=9 if x>9 E. f(x)=x−91 F. f(x)=⎩⎨⎧2x02x−36 if x<0 if x=9 if x>9
f(x)={5x,x2,x≤3x>3. Find the indlcated one-sided limis of f, and determine the continuly of f at the indicated point. You should also sketch a graph of y=f(x), including hollow and solid circlos in the appropriate places: NOTE: Type DNE A a limit does not exist. limx→3−f(x)=limx→1ff(x)=limx→3f(x)=f(3)= is f continuous at 3? (YESNO)