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The Limit of a Function

Numerical Limit:

1. Guess the value of lim   Solution

2. Guess the value of \displaystyle{\lim_{x \to -3}   \frac{5x+1}{2x-1}}    Solution

3. Guess the value of \displaystyle{\lim_{x \to 0}   \frac{\sin x }{x }}     Solution

 

Graphical Limits:

1. For the function f whose graph is given, state the value of each, if it exists.

graph

(a) \displaystyle{\lim_{x \to -2^+}   f(x) }   (b) \displaystyle{\lim_{x \to -2^-}   f(x) }     (c) \displaystyle{\lim_{x \to -2}   f(x) }     (d) \displaystyle{\lim_{x \to -3}   f(x) }     (e) \displaystyle{\lim_{x \to 4^-}   f(x) }   (f) \displaystyle{\lim_{x \to 4^+}   f(x) }     (g) \displaystyle{\lim_{x \to 4}   f(x) }    (h) \displaystyle{\lim_{x \to 2}   f(x) }   (i)  \displaystyle{f(-3)}   Solution

2. Sketch the graph of an example of a function f that satisfies all of the given conditions.

I. \displaystyle \lim_{x \to 0^-} f(x)=-2, \, \lim_{x \to 0^+}f(x)=1, \, f(0)=3    Solution

II. \displaystyle \lim_{x \to 5^+} f(x)=4, \, \lim_{x \to 5^-}f(x)=3, \, \lim_{x \to -2}f(x)=5, \, f(5)=4, \, f(-2)=-1   Solution

III. \displaystyle{\lim_{x \to 0} f(x)=1, \, f(0)=2, \, \lim_{x \to 3^+} f(x)=\infty, \,\lim_{x \to 3^-} f(x)=-\infty},\, f(-1)=0    Solution

 

Infinite Limits: Determine the infinite limit.

(1) (a) \displaystyle{\lim_{x \to 3^+} \frac{x+1}{x-3}}   (b) \displaystyle{\lim_{x \to 2^-} \frac{1}{x-2}}   solution 

(2) Evaluate the one-sided limits for \, \displaystyle{f(x)=\frac{2}{x^3-1}}.    (a) \displaystyle{\lim_{x\to1^-} \, f(x)}    (b) \displaystyle{\lim_{x\to1^+} \, f(x)}     solution

 

 

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