Section 2.2: The Limit of a Function

Numerical Limits

1. Guess the value of the limit: \(\displaystyle{\lim_{x \to 2} (3x-5)}\) View Solution
2. Guess the value of the limit: \(\displaystyle{\lim_{x \to -3} \frac{5x+1}{2x-1}}\) View Solution
3. Guess the value of the limit: \(\displaystyle{\lim_{x \to 0} \frac{\sin x }{x }}\) View Solution

Graphical Limits

1. For the function \( f \) whose graph is given below, state the value of each, if it exists. View Solution (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) \(f(-3)\)
2. Sketch the graph of an example of a function \( f\) that satisfies all of the given conditions. I. \( \displaystyle \lim_{x \to 0^-} f(x)=-2, \, \lim_{x \to 0^+}f(x)=1, \, f(0)=3 \) View Solution II. \( \displaystyle \lim_{x \to 5^+} f(x)=4, \, \lim_{x \to 5^-}f(x)=3, \, \lim_{x \to -2}f(x)=5, \, f(5)=4, \, f(-2)=-1 \) View Solution III. \( \displaystyle{\lim_{x \to 0} f(x)=1, \, f(0)=2, \, \lim_{x \to 3^+} f(x)=\infty, \,\lim_{x \to 3^-} f(x)=-\infty},\, f(-1)=0 \) View Solution

Infinite Limits

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