When Does a Limit Not Exist

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.

Calculus in Maths
Differential Calculus
Limit in Calculus
Limits, Continuity and Differentiability