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One-sided limits of piecewise functions practice problems

Left- and right-hand limits of piecewise functions, and when the two-sided limit exists. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Let \( \displaystyle f(x) = \begin{cases} 3 - x & x < 0 \\ -2 & x = 0 \\ 2 x^{2} + 2 x & x > 0 \end{cases} \). Find \( \displaystyle \lim_{x\to 0^-} f(x) \), \( \displaystyle \lim_{x\to 0^+} f(x) \), \( \displaystyle \lim_{x\to 0} f(x) \) and \( \displaystyle f(0) \).

Problem 1.261easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 3 x^{2} - 2 x + 1 & x < 0 \\ -2 & x = 0 \\ - 3 x^{2} - x & x > 0 \end{cases} \). Find \( \displaystyle \lim_{x\to 0^-} f(x) \), \( \displaystyle \lim_{x\to 0^+} f(x) \), \( \displaystyle \lim_{x\to 0} f(x) \) and \( \displaystyle f(0) \).

Problem 1.262easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 3 x - 4 & x < 0 \\ 5 & x = 0 \\ - 2 x^{2} + x - 4 & x > 0 \end{cases} \). Find \( \displaystyle \lim_{x\to 0^-} f(x) \), \( \displaystyle \lim_{x\to 0^+} f(x) \), \( \displaystyle \lim_{x\to 0} f(x) \) and \( \displaystyle f(0) \).

Problem 1.263easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 3 x^{2} - x + 1 & x < 3 \\ -4 & x = 3 \\ - 3 x^{2} - x + 1 & x > 3 \end{cases} \). Find \( \displaystyle \lim_{x\to 3^-} f(x) \), \( \displaystyle \lim_{x\to 3^+} f(x) \), \( \displaystyle \lim_{x\to 3} f(x) \) and \( \displaystyle f(3) \).

Problem 1.264easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} 2 x - 3 & x < -2 \\ -1 & x = -2 \\ 4 x + 2 & x > -2 \end{cases} \). Find \( \displaystyle \lim_{x\to -2^-} f(x) \), \( \displaystyle \lim_{x\to -2^+} f(x) \), \( \displaystyle \lim_{x\to -2} f(x) \) and \( \displaystyle f(-2) \).

Problem 1.265easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 4 x - 4 & x < -1 \\ 1 & x = -1 \\ - 2 x^{2} - 2 x & x > -1 \end{cases} \). Find \( \displaystyle \lim_{x\to -1^-} f(x) \), \( \displaystyle \lim_{x\to -1^+} f(x) \), \( \displaystyle \lim_{x\to -1} f(x) \) and \( \displaystyle f(-1) \).

Problem 1.266easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} x + 4 & x < 2 \\ -1 & x = 2 \\ 12 - 3 x & x > 2 \end{cases} \). Find \( \displaystyle \lim_{x\to 2^-} f(x) \), \( \displaystyle \lim_{x\to 2^+} f(x) \), \( \displaystyle \lim_{x\to 2} f(x) \) and \( \displaystyle f(2) \).

Problem 1.267easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 3 x^{2} + x + 2 & x < 3 \\ 0 & x = 3 \\ - x^{2} - x + 3 & x > 3 \end{cases} \). Find \( \displaystyle \lim_{x\to 3^-} f(x) \), \( \displaystyle \lim_{x\to 3^+} f(x) \), \( \displaystyle \lim_{x\to 3} f(x) \) and \( \displaystyle f(3) \).

Problem 1.268easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} - 3 x^{2} + 2 x & x < -2 \\ 2 & x = -2 \\ x^{2} + 2 x - 16 & x > -2 \end{cases} \). Find \( \displaystyle \lim_{x\to -2^-} f(x) \), \( \displaystyle \lim_{x\to -2^+} f(x) \), \( \displaystyle \lim_{x\to -2} f(x) \) and \( \displaystyle f(-2) \).

Problem 1.269easy✓ Nihil obstat

Let \( \displaystyle f(x) = \begin{cases} 3 - 4 x & x < 0 \\ 1 & x = 0 \\ 3 x^{2} - 2 x + 3 & x > 0 \end{cases} \). Find \( \displaystyle \lim_{x\to 0^-} f(x) \), \( \displaystyle \lim_{x\to 0^+} f(x) \), \( \displaystyle \lim_{x\to 0} f(x) \) and \( \displaystyle f(0) \).

Problem 1.270easy✓ Nihil obstat