Limits of Constant and Linear Functions

The limits of constant and linear functions are are presented with examples and detailed solutions. Exercises are also included with their answers.

Limits of Constant and Linear Functions

We present the limits of constant and linear functions.
1. For \( f(x) = c \) , where \( c \) is a constant ,

\(\displaystyle \lim_{x \to a} f(x) = c \)
\(\displaystyle \lim_{x \to +\infty} f(x) = c \)
\(\displaystyle \lim_{x \to -\infty} f(x) = c \)

2. For \( f(x) = x \) , basic linear function

\(\displaystyle \lim_{x \to a} f(x) = a \)
\(\displaystyle \lim_{x \to +\infty} f(x) = +\infty \)
\(\displaystyle \lim_{x \to -\infty} f(x) = -\infty \)

Examples with Detailed Solutions

Example 1

Find the following limits.

\(\displaystyle \lim_{x \to 6} (-2) \)
\(\displaystyle \lim_{x \to -\infty} (0) \)
\(\displaystyle \lim_{x \to +\infty} (x) \)

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 \[ \lim_{x\to\ -\infty} (0) = 0 \] 3. 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. \( \lim_{x\to\ 1/2} (x) = 1/2 \)

Limits of Constant and Linear Functions