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.

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