Limits, L'Hôpital's Rule, and Continuity

1. The Core Idea: What is a Limit?

In calculus, we are often interested in a function's behavior near a point, not just at the point. A limit describes the value a function "approaches" as its input gets arbitrarily close to a certain value. This is crucial when a function is undefined at a specific point, yet we can still describe its behavior around that point.

To build a solid intuition, let's start with a video that explains the concept of a limit, the difference between a function's value and its limit, and the idea of approaching from the left (Left-Hand Limit, LHL) and the right (Right-Hand Limit, RHL).

As the video established, the limit of $f(x)$ as $x$ approaches $c$ exists if and only if both the left-hand and right-hand limits exist and are equal:

If this condition holds, we say ${\lim }_{x\to c}f(x)=L$.

2. Evaluating Limits: The Engineer's Toolkit

For GATE, you need efficient methods to calculate limits. The first step is always direct substitution. If plugging the value $c$ into $f(x)$ gives a finite number, that's your answer. However, you'll often encounter indeterminate forms. These are expressions where the result is not immediately known, such as:

Our primary weapon against the $\frac{0}{0}$ and $\frac{\infty }{\infty }$ forms is L'Hôpital's Rule.

L'Hôpital's Rule

Statement: If ${\lim }_{x\to c}f(x)$ results in the indeterminate form $\frac{0}{0}$ or $\frac{\infty }{\infty }$, then:

...provided the limit on the right-hand side exists.

Crucial Caveat: Always check that you have a $\frac{0}{0}$ or $\frac{\infty }{\infty }$ form before applying the rule. Applying it otherwise will lead to an incorrect answer.

Let's watch a series of examples showing L'Hôpital's rule in action.

Handling Other Indeterminate Forms

What about forms like $1^{\infty }$ or $\infty -\infty$? The strategy is to algebraically manipulate them into the $\frac{0}{0}$ or $\frac{\infty }{\infty }$ form so you can apply L'Hôpital's rule.

Let's see how this is done with two common types from GATE.

A common GATE question pattern involves a limit that has a finite value, but contains an unknown constant. To solve it, you often have to deduce that the limit must be in an indeterminate form for a finite value to exist.

3. Continuity: Is the Path Unbroken?

The concept of a limit leads directly to the definition of continuity. Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. Formally, a function $f(x)$ is continuous at a point $x=c$ if it satisfies three conditions:

1. $f(c)$ is defined: The point must exist on the graph.
2. ${\lim }_{x\to c}f(x)$ exists: The function must be approaching a single finite value from both sides (LHL = RHL).
3. ${\lim }_{x\to c}f(x)=f(c)$: The value the function approaches must be the same as the function's actual value at that point.

If any of these conditions fail, the function is discontinuous at $x=c$. For example, in the function $f(x)=\frac{x^2-1}{x-1}$, the limit at $x=1$ is 2, but $f(1)$ is undefined (violates condition 1). This is a "removable discontinuity" because we could define $f(1)=2$ to make it continuous.

4. Differentiability: Is the Path Smooth?

Differentiability is a stricter condition than continuity. A function is differentiable at a point if its graph is not only unbroken but also "smooth," with no sharp corners or cusps.

Formally, a function $f(x)$ is differentiable at a point $x=c$ if the following limit exists:

This limit represents the instantaneous rate of change, or the slope of the tangent line to the graph at that point.

For the limit to exist, the left-hand and right-hand limits must be equal:

• Left-Hand Derivative (LHD): ${\lim }_{h\to 0^{-}}\frac{f(c+h)-f(c)}{h}$
• Right-Hand Derivative (RHD): ${\lim }_{h\to 0^{+}}\frac{f(c+h)-f(c)}{h}$

Differentiability at $x=c$ requires LHD = RHD.

A classic example that distinguishes these concepts is the absolute value function, $f(x)=|x|$, at $x=0$.

• Continuity: It's continuous. $f(0)=0$, and ${\lim }_{x\to 0}|x|=0$. All three conditions hold.
• Differentiability: It's not differentiable.
  • RHD at 0: ${\lim }_{h\to 0^{+}}\frac{|0+h|-|0|}{h}={\lim }_{h\to 0^{+}}\frac{h}{h}=1$
  • LHD at 0: ${\lim }_{h\to 0^{-}}\frac{|0+h|-|0|}{h}={\lim }_{h\to 0^{-}}\frac{-h}{h}=-1$
    Since LHD $\ne$ RHD, the function has a "sharp corner" at $x=0$ and is not differentiable there.

This leads to a fundamental theorem:

If a function is differentiable at a point, it must be continuous at that point.
The converse is not true (as shown by $f(x)=|x|$).

Conclusion

In this lesson, we laid the groundwork for calculus by exploring limits, continuity, and differentiability. You've learned how to analyze a function's behavior near a point and equipped yourself with a powerful tool for evaluating tricky limits.

Key Takeaways:

• A limit exists if and only if the Left-Hand Limit (LHL) equals the Right-Hand Limit (RHL).
• L'Hôpital's Rule is your go-to method for indeterminate forms $\frac{0}{0}$ and $\frac{\infty }{\infty }$. Remember to verify the form before applying it.
• Other indeterminate forms ($1^{\infty }$, $\infty -\infty$, etc.) must be algebraically converted to $\frac{0}{0}$ or $\frac{\infty }{\infty }$ before using L'Hôpital's rule.
• Continuity at a point requires three conditions: the function is defined, the limit exists, and the limit equals the function's value.
• Differentiability at a point requires the Left-Hand Derivative to equal the Right-Hand Derivative. It is a stronger condition than continuity.

In our next lesson, we will apply these concepts. We will use the derivative, which we defined today as a limit, to find the maxima and minima of functions and explore important results like the Mean Value Theorems.