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6.* Find an example of a function such that the limit exists at every x, but that has an in nite number of discontinuities. (You can describe the function and/or write a formula down and/or draw a graph.) PARTIAL ANSWERS: 1. (a) x = 0;3 (b) x = 2;0;1 2. (a) R (b) Rnf 1=2;2g (c) (1 ;5] (d) ( 3;2)[( 2;2)[(2;4) 3.
Limits are a very powerful tool in mathematics and are used throughout calculus and beyond. The key idea is that a limit is what I like to call a \behavior operator". A limit will tell you the behavior of a function nearby a point.
Limits. Basic. Divergence. 1.\:\:\lim _ {x\to 0} (\frac {1} {x}) 2.\:\:\lim _ {x\to 5} (\frac {10} {x-5}) 3.\:\:\lim _ {x\to 1} (\frac {x} {x-1}) 4.\:\:\lim _ {x\to -2} (\frac {1} {x+2}) 5.\:\:\lim _ {x\to 5} (\frac {x} {x^2-25}) 6.\:\:\lim _ {x\to 2}\frac {|x-2|} {x-2}
The limits are defined as the value that the function approaches as it goes to an x value. Using this definition, it is possible to find the value of the limits given a graph. A few examples are below: . In general, you can see that these limits are equal to the value of the function. This is true if the function is continuous. Continuity .
Definition of a Limit If f (x) gets arbitrarily close to a single number L as x approaches c, we lim f(x)=L then Note from the definition: 1) The limit is unique if it exists. (limit from the left = limit from the fight) 2) The limit does not depend on the actual value of f (x) at c. Instead, it is determined by values of f (x) when x is near c
CALCULUS AB WORKSHEET 1 ON LIMITS. Work the following on notebook paper. No calculator. 1. The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. ( a ) lim ⎡ ⎣ f ( x g ( x. x → 2. ( ) c lim ⎡ f x g x ⎤. x → 0 ⎣ ( ) ( ) ⎦. ( b ) lim f. → 1 ⎡ ⎣ ( x g ( x ) ⎤ ⎦. ( x ) lim ( )
Here are some examples: 3.2. Example. The function f(x) = (x3 1)=(x 1) is at rst not de ned at x = 1. But for x close to 1, nothing really bad happens. We can evaluate the function at points closer and closer to 1 and get closer and closer to 3. We say limx!1 f(x) = 3.
