One-Sided Limits — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}2x+1,& x<0\\ x^2+3,& x\ge 0\end{array}\underset{x\to 0^{-}}{\lim }f(x)$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{A left-hand limit follows the left branch.}$$
   2. $$\underset{x\to 0^{-}}{\lim }2x+1=1$$
2. 2.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}x+4,& x<1\\ 3x-1,& x\ge 1\end{array}\underset{x\to 1^{+}}{\lim }f(x)$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{A right-hand limit follows the right branch.}$$
   2. $$\underset{x\to 1^{+}}{\lim }3x-1=2$$
3. 3.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}{x}^2-1,& x<2\\ 5-x,& x\ge 2\end{array}\underset{x\to 2^{-}}{\lim }f(x)$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{A left-hand limit follows the left branch.}$$
   2. $$\underset{x\to 2^{-}}{\lim }{x}^2-1=3$$
4. 4.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}2-x,& x<-1\\ x^2+2,& x\ge -1\end{array}\underset{x\to -1^{+}}{\lim }f(x)$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{A right-hand limit follows the right branch.}$$
   2. $$\underset{x\to -1^{+}}{\lim }{x}^2+2=3$$
5. 5.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}x-2,& x<3\\ 2x+5,& x\ge 3\end{array}\underset{x\to 3^{+}}{\lim }f(x)$$

   Answer: $11$

   Step-by-step solution

   1. $$\text{A right-hand limit follows the right branch.}$$
   2. $$\underset{x\to 3^{+}}{\lim }2x+5=11$$
6. 6.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}4x-3,& x<2\\ x^2,& x\ge 2\end{array}\underset{x\to 2^{-}}{\lim }f(x)$$

   Answer: $5$

   Step-by-step solution

   1. $$\text{A left-hand limit follows the left branch.}$$
   2. $$\underset{x\to 2^{-}}{\lim }4x-3=5$$
7. 7.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}x+5,& x<0\\ 3x+1,& x\ge 0\end{array}\underset{x\to 0^{+}}{\lim }f(x)$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{A right-hand limit follows the right branch.}$$
   2. $$\underset{x\to 0^{+}}{\lim }3x+1=1$$
8. 8.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}2x-1,& x<1\\ x^2+2,& x\ge 1\end{array}\underset{x\to 1^{-}}{\lim }f(x)$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{A left-hand limit follows the left branch.}$$
   2. $$\underset{x\to 1^{-}}{\lim }2x-1=1$$
9. 9.

   Evaluate the one-sided limit.

   $$f(x)=\{\begin{array}{ll}x+3,& x<-2\\ 2x^2,& x\ge -2\end{array}\underset{x\to -2^{+}}{\lim }f(x)$$

   Answer: $8$

   Step-by-step solution

   1. $$\text{A right-hand limit follows the right branch.}$$
   2. $$\underset{x\to -2^{+}}{\lim }2x^2=8$$
10. 10.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}{x}^2-5,& x<4\\ 3x,& x\ge 4\end{array}\underset{x\to 4^{-}}{\lim }f(x)$$

    Answer: $11$

    Step-by-step solution

    1. $$\text{A left-hand limit follows the left branch.}$$
    2. $$\underset{x\to 4^{-}}{\lim }{x}^2-5=11$$
11. 11.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}5-x,& x<2\\ x^2+1,& x\ge 2\end{array}\underset{x\to 2^{+}}{\lim }f(x)$$

    Answer: $5$

    Step-by-step solution

    1. $$\text{A right-hand limit follows the right branch.}$$
    2. $$\underset{x\to 2^{+}}{\lim }{x}^2+1=5$$
12. 12.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}3x+2,& x<-1\\ x-4,& x\ge -1\end{array}\underset{x\to -1^{-}}{\lim }f(x)$$

    Answer: $-1$

    Step-by-step solution

    1. $$\text{A left-hand limit follows the left branch.}$$
    2. $$\underset{x\to -1^{-}}{\lim }3x+2=-1$$
13. 13.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}{x}^2+x,& x<3\\ 2-x,& x\ge 3\end{array}\underset{x\to 3^{+}}{\lim }f(x)$$

    Answer: $-1$

    Step-by-step solution

    1. $$\text{A right-hand limit follows the right branch.}$$
    2. $$\underset{x\to 3^{+}}{\lim }2-x=-1$$
14. 14.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}4-x^2,& x<1\\ 5x+2,& x\ge 1\end{array}\underset{x\to 1^{-}}{\lim }f(x)$$

    Answer: $3$

    Step-by-step solution

    1. $$\text{A left-hand limit follows the left branch.}$$
    2. $$\underset{x\to 1^{-}}{\lim }4-x^2=3$$
15. 15.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}2-3x,& x<0\\ x^2+6,& x\ge 0\end{array}\underset{x\to 0^{+}}{\lim }f(x)$$

    Answer: $6$

    Step-by-step solution

    1. $$\text{A right-hand limit follows the right branch.}$$
    2. $$\underset{x\to 0^{+}}{\lim }{x}^2+6=6$$
16. 16.

    Evaluate the one-sided limit.

    $$f(x)=\{\begin{array}{ll}x+1,& x<5\\ x^2-10,& x\ge 5\end{array}\underset{x\to 5^{-}}{\lim }f(x)$$

    Answer: $6$

    Step-by-step solution

    1. $$\text{A left-hand limit follows the left branch.}$$
    2. $$\underset{x\to 5^{-}}{\lim }x+1=6$$

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