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