What is the Limit of a Function

The limit of a function is a fundamental concept in calculus and mathematical analysis, describing the behavior of a function as its input approaches a particular value. Simply put, a function f(x) has a limit L at x = a if, as x gets closer to a, the values of f(x) approach L.

In this article, we will discuss Limit of a Function in detail.

Definition of a Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a particular point. Formally, if f(x) is a function and c is a point in its domain the limit of f(x) as the x approaches c is denoted as:

lim_x→cf(x)=L

where L is the value that f(x) gets closer to as the x gets closer to c. This can be interpreted as the function f(x) approaching L from both the sides of the c.

Epsilon-Dealt Definition of Limit

For a given function f(x), the limit lim_⁡x→cf(x) = L holds if, for every small positive number ϵ, there exists a corresponding small number δ such that whenever x is within δ-distance of c (but not equal to c), the value of f(x) is within ϵ-distance of L.

In formal notation, this is expressed as:

∀ ϵ > 0, ∃ δ > 0 such that if 0 < ∣x − c∣ < δ, then ∣f(x) − L∣ < ϵ.

Types of Limits

Finite Limits: If the function approaches a finite value as the variable approaches a certain point.

Infinite Limits: If the function grows without the bound as the variable approaches a particular value.

One-Sided Limits OR Left-hand and Right-hand Limits:

• Left-hand limit: The value that f(x) approaches as x approaches c from left (denoted as \lim\limits_{x \to c^-} f(x)).
• Right-hand limit: The value that f(x) approaches as x approaches c from right (denoted as \lim\limits_{x \to c^+} f(x)).

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

\lim\limits_{x \to c^-} f(x) = \lim\limits_{x \to c^+} f(x) = L

If these two are not equal the limit at c does not exist.

Limits at Infinity: The Limits where x approaches infinity or negative infinity. These describe the behavior of the function as the variable grows indefinitely.

Explanation of Limits using Table

let’s consider the function f(x)=x² and evaluate its limit as x approaches 2.

We want to calculate lim⁡_x→2f(x) = lim⁡_x→2x².

Now, let’s create a table with values of x getting closer to 2 from both sides (left and right) and observe the corresponding f(x) = x² values:

Asx approaches 2 from both the left (values less than 2) and the right (values greater than 2), the values of f(x) get closer and closer to 4. This shows that lim⁡_x→2x² = 4.

Thus, the limit of f(x) = x² as x approaches 2 is 4, confirmed through this table.

Laws of Limits

The Limits follow specific rules that make solving limit problems easier. Here are some fundamental laws of limits:

Sum Law

The limit of a sum of the functions is equal to the sum of their limits.

\lim\limits_{x \to c} [f(x) + g(x)] = \lim\limits_{x \to c} f(x) + \lim\limits_{x \to c} g(x)

Difference Law

The limit of the difference of the two functions is equal to the difference of their limits.

\lim\limits_{x \to c} [f(x) - g(x)] = \lim\limits_{x \to c} f(x) - \lim\limits_{x \to c} g(x)

Product Law

The limit of the product of the two functions is the product of their limits.

\lim\limits_{x \to c} [f(x) \cdot g(x)] = \lim\limits_{x \to c} f(x) \cdot \lim\limits_{x \to c} g(x)

Quotient Law

The limit of a quotient of the two functions is the quotient of their limits provided the denominator's limit is non-zero.

\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}, \text{ if } \lim\limits_{x \to c} g(x) \neq 0

Constant Multiple Law

The limit of a constant multiplied by the function is the constant multiplied by the limit of the function.

\lim\limits_{x \to c} [k \cdot f(x)] = k \cdot \lim\limits_{x \to c} f(x)

Power Law

The limit of a function raised to the power n is the limit of the function raised to that power.

\lim\limits_{x \to c} [f(x)]^n = \left(\lim\limits_{x \to c} f(x)\right)^n

Methods to Calculate Limits

Some of the common methods to calculate limits are:

• Direct Substitution: For continuous functions directly substitute the value of the x into the function.
• Factoring: The Factorize the function to the simplify expressions especially for the rational functions.
• Rationalizing: For functions with square roots rationalize the numerator or denominator to the resolve indeterminate forms.
• L'Hospital Rule: For indeterminate forms like (0\0) or (∞\∞) differentiate the numerator and denominator and then evaluate the limit.
• Graphical Method: Analyze the graph of the function to the estimate the limit.

Applications of Limits

Limit is the foundation of calculus and some of the common application of limits in calculus are:

• Derivatives: The Limits are used to define derivatives which measure the rate of the change of a function.
• Integrals: The Limits are fundamental in defining the integrals which represent the area under a curve.
• Continuity: The concept of limits is used to the determine if a function is continuous at a point.
• Asymptotic Behavior: The Limits help in understanding the behavior of the functions as they approach infinity or other critical points.

Examples with Solutions

Example 1: Limit of a Polynomial Function

Problem: Find the limit of the function f(x) = 2x³ - 4x + 1 as x approaches 3.

Solution:

To find the limit substitute x = 3 into the function:

\lim_{x \to 3} (2x^3 - 4x + 1) = 2(3)^3 - 4(3) + 1 = 2(27) - 12 + 1 = 54 - 12 + 1 = 43

So,

\lim_{x \to 3} (2x^3 - 4x + 1) = 43

Example 2: Limit of a Rational Function

Problem: Find the limit of g(x) = \frac{x^2 - 4}{x - 2} as x approaches 2.

Solution:

Direct substitution results in \frac{0}{0} so we need to the simplify:

Factor the numerator:

x² - 4 = (x - 2)(x + 2)

Thus, g(x) = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \text{ for } x \neq 2

So, \lim_{x \to 2} (x + 2) = 2 + 2 = 4

Thus,\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4

Example 3: Limit of a Function with Square Roots

Problem: Find the limit of the h(x) = \frac{\sqrt{x + 4} - 2}{x - 0} as x approaches 0.

Solution:

Multiply numerator and denominator by the conjugate of the numerator:

\frac{\sqrt{x + 4} - 2}{x} \cdot \frac{\sqrt{x + 4} + 2}{\sqrt{x + 4} + 2} = \frac{(\sqrt{x + 4})^2 - 2^2}{x(\sqrt{x + 4} + 2)} = \frac{x + 4 - 4}{x(\sqrt{x + 4} + 2)} = \frac{x}{x(\sqrt{x + 4} + 2)} = \frac{1}{\sqrt{x + 4} + 2}

Now,

\lim_{x \to 0} \frac{1}{\sqrt{x + 4} + 2} = \frac{1}{\sqrt{0 + 4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}

So,

\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x} = \frac{1}{4}

Example 4: Limit Using L'Hospital Rule

Problem: Find the limit of \frac{e^x - 1}{x} as x approaches 0.

Solution:

Direct substitution results in \frac{0}{0}. Use L'Hospital Rule by the differentiating the numerator and denominator:

\lim_{x \to 0} \frac{e^x - 1}{x} = \lim_{x \to 0} \frac{e^x}{1} = e^0 = 1

So,

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Example 5: Limit of a Trigonometric Function

Problem: Find the limit of \sin(x) as the x approaches \frac{\pi}{2} .

Solution:

Direct substitution:

\lim_{x \to \frac{\pi}{2}} \sin(x) = \sin\left(\frac{\pi}{2}\right) = 1

So,

\lim_{x \to \frac{\pi}{2}} \sin(x) = 1

Practical Questions

1. Find \lim_{x \to 1} (x^2 + 3x - 4) .
2. Evaluate \lim_{x \to -1} \frac{x + 1}{x^2 + 1}.
3. Determine \lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}.
4. Calculate \lim_{x \to 0} \frac{\ln(x + 1)}{x}.
5. Find \lim_{x \to \infty} \frac{3x^2 - 5x + 2}{2x^2 + x - 1}.
6. Evaluate \lim_{x \to \frac{\pi}{4}} \tan(x).
7. Determine \lim_{x \to 0} \frac{1 - \cos(x)}{x^2}.
8. Find \lim_{x \to \infty} \frac{5x^3 - 2x}{4x^3 + 3}.
9. Calculate \lim_{x \to -2} \frac{x + 2}{x^3 + 8}.
10. Evaluate \lim_{x \to 3} \frac{x^2 - 9}{x - 3}.

Answer Key

1. 0
2. 0
3. 4
4. 1
5. 3/2
6. 1
7. 1/2
8. 5/4
9. 1/12
10. 6

Conclusion

Understanding limits is essential for mastering calculus and mathematical analysis. They provide a foundational tool for the analyzing and interpreting the behavior of functions leading to the deeper insights into calculus concepts such as the derivatives and integrals. Mastery of limits allows for the precise mathematical reasoning and problem-solving across various fields.

Read More,

• Calculus
• Limits in Calculus
• Continuity and Differentiability
• Derivatives
• Definite Integral