Limits of Basic and Common Functions

The limits of two basic and common functions are are presented with examples and detailed solutions. Exercises are also included with their answers.

Limits of Basic Functions


We present the limits of some basic functions.
1. For f(x) = c where c is a constant ,
  • a.
    \lim_{x\to\ a} f(x) = c

  • b.
    \lim_{x\to\ +\infty} f(x) = c

  • c.
    \lim_{x\to\ -\infty} f(x) = c

2. For f(x) = x,

  • a.
    \lim_{x\to\ a} f(x) = a

  • b.
    \lim_{x\to\ +\infty} f(x) = +\infty

  • c.
    \lim_{x\to\ -\infty} f(x) = -\infty

Example 1: Find the following limits.
1.

\lim_{x\to\ 6} (-2)

2.
\lim_{x\to\ -\infty} (0)

3.
\lim_{x\to\ +\infty} (x)

Solutions to Example 1:
1. The function is constant and equal to -2, we apply 1.a above, hence

\lim_{x\to\ 6} (-2) = -2

2.
The function is constant and equal to 0, we apply 1.c above, hence
\lim_{x\to\ -\infty} (0) = 0

2.
The function is equal to x, we apply 2.b above, hence
\lim_{x\to\ +\infty} (x) = +\infty

Exercises: Find the following limits.
1.

\lim_{x\to\ -\infty} (6)

2.
\lim_{x\to\ -5} (0)

3.
\lim_{x\to\ -\infty} x

4.
\lim_{x\to\ 1/2} (x)

Answers to Above Exercises:
1.

\lim_{x\to\ -\infty} (6) = 6

2.
\lim_{x\to\ -5} (0) = 0

3.
\lim_{x\to\ -\infty} x = -\infty

4.
\lim_{x\to\ 1/2} (x) = 1/2


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