The limits of constant and linear functions are presented with examples and detailed solutions. Exercises are also included with their answers.
In what follows, \( C \) represents the constant of integration where applicable.
Limits of Constant and Linear Functions
We present the fundamental limit rules for 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 \( f(x) = x \), the 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
Click on each example to view its detailed step-by-step solution.
Example 1
Find the following limits:
- \( \displaystyle \lim_{x \to 6} (-2) \)
- \( \displaystyle \lim_{x \to -\infty} (0) \)
- \( \displaystyle \lim_{x \to +\infty} (x) \)
Show Solutions to Example 1
1. The function is constant and equal to \( -2 \). Applying rule 1.a above:
\[ \lim_{x \to 6} (-2) = -2 \]2. The function is constant and equal to \( 0 \). Applying rule 1.c above:
\[ \lim_{x \to -\infty} (0) = 0 \]3. The function is equal to \( x \). Applying rule 2.b above:
\[ \lim_{x \to +\infty} (x) = +\infty \]Exercises
Find the following limits. Click each exercise to check your answers.
Exercise 1
\( \displaystyle \lim_{x \to -\infty} (6) \)
Show Answer
Exercise 2
\( \displaystyle \lim_{x \to -5} (0) \)
Show Answer
Exercise 3
\( \displaystyle \lim_{x \to -\infty} x \)
Show Answer
Exercise 4
\( \displaystyle \lim_{x \to 1/2} (x) \)