When Does a Limit Not Exist (original) (raw)
Last Updated : 8 Jun, 2026
A limit exists only when the function approaches one unique value from both sides of a point. If this condition fails, the limit does not exist.
Consider the function f(x)=\frac{1}{x}
As x→0:
- From the left side (x \to 0^-),f(x)→−∞
- From the right side (x \to 0^+),f(x)→+∞
Since the values approached from the two sides are different, the function does not approach a single value. Therefore, \lim_{x \to 0} \frac{1}{x} does not exist.
Cases When a Limit Does Not Exist
There are many possible cases when the limit does not exist:
**1. Different Left-hand and Right-hand Limits
If the limit from the left-hand side (as x approaches a point from values smaller than the target point) and the limit from the right-hand side (as x approaches from values greater than the target point) are not equal, the overall limit does not exist.
- **Example: lim_{x→0^-}\frac{1}{x}= −∞\ \text{and} \ lim_{x→0^+}\frac{1}{x}= +∞. Since these limits are not the same, the limit at x = 0 does not exist.
**2. Unbounded Behavior (Approaching Infinity)
If the function grows infinitely large (positive or negative) as it approaches the target point, the limit does not exist in a real sense, even though we can describe it using infinity.
- **Example:\lim_{x \to 0} \frac{1}{x^2} = +\infty. Since the function becomes unbounded, we say the limit does not exist.
**3. Oscillatory Behavior
If the function oscillates between two or more values as it approaches the target point without settling on a single value, the limit does not exist.
- **Example:\lim_{x \to 0} \sin\left(\frac{1}{x}\right). This function oscillates infinitely between -1 and 1 as x approaches 0, so the limit does not exist.
**4. Discontinuities
If the function has a jump or a gap at the point of interest, the limit may not exist.
- **Example: For the step function f(x)f(x) = \begin{cases} 1, & \text{if } x < 0 \\ 2, & \text{if } x \geq 0 \end{cases}, the left-hand limit is 1 and the right-hand limit is 2, so the limit does not exist at x = 0.
Solved Examples
**Example 1: Evaluate: \lim_{x \to 0} \frac{|x|}{x}
The function \frac{|x|}{x} behaves differently when x approaches 0 from the left and right:
- When x > 0 (approaching from the right): |x| = x, so \frac{|x|}{x} = 1.
- When x < 0 (approaching from the left): |x| = -x, so \frac{|x|}{x} = -1.
Since the left-hand limit is -1 and the right-hand limit is 1, the overall limit does not exist.
**Example 2: Evaluate \lim_{x\lim_{x \to 0} \frac{1}{x^2}
As x approaches 0 from either side, 1/x2 becomes very large because x2 is always positive and gets smaller as x → 0.
- When x → 0+: \frac{1}{x^2} \to +\infty
- When x → 0-: \frac{1}{x^2} \to +\infty
Since the function becomes unbounded as x \to 0, the limit does not exist in a finite sense.
**Example 3: Evaluate \lim_{x \to 0} \sin\left(\frac{1}{x}\right)
As x → 0, 1/x grows larger and larger, causing \sin\left(\frac{1}{x}\right) to oscillate rapidly between -1 and 1.
The function does not approach any single value as x \to 0 because of this oscillatory behavior. Thus, the limit does not exist.
**Example 4: Evaluate \lim_{xEvaluate: \lim_{x \to 0} f(x), where \ f(x) = \begin{cases}1, & x < 0 \\2, & x \geq 0\end{cases}
- The left-hand limit as x → 0- is .\lim_{x \to 0^-} f(x) = 1
- The right-hand limit as x → 0+ is \lim_{x \to 0^+} f(x) = 2.
Since the left-hand limit and right-hand limit are not equal, the limit does not exist.
**Example 5: Evaluate \lim_{xEvaluate: \lim_{x \to 1} f(x), where \ f(x) = \begin{cases}x^2, & x < 1 \\2x, & x \geq 1\end{cases}
- The left-hand limit as x → 1- is \lim_{x \to 1^-} x^2 = 1^2 = 1.
- The right-hand limit as x → 1+ is \lim_{x \to 1^+} 2x = 2(1) = 2.
Since the left-hand limit and right-hand limit are not equal, the limit does not exist.