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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.

(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