The Limit of a Function

Numerical Limit:

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

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