Limit of a Function: Definition, Properties, and Calculations -

Limit of a Function: Definition, Properties, and Calculations

Limit of a Function: Definition, Properties, and Calculations
Limit of a Function: Definition, Properties, and Calculations

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is a fundamental concept in calculus and is used to define derivatives and integrals. To explain limits, we can think of a function as a machine that takes inputs and produces outputs.

Limits describe what happens to the output of a function as we change the input. Specifically, we are interested in what happens to the output as the input gets closer and closer to a certain value. In this article, we will discuss the definition of limit, special rules for limits, and some examples.

Definition of limit

Limits in math are unique real numbers. Let us suppose a real-valued function f and the real number c. The limit is normally read as “the limit of “f of x”, as the variable x approaches c equal to L. The lim represents the Limit and the function f(x) approaches the limit L as x approached c is described by the right arrow.

For instance, consider the function f(x) = 1/x. As x gets closer and closer to 0, the value of f(x) gets larger and larger (in absolute value). The limit of f(x) as x approaches 0 is infinity, which we write as:

Limx→0 f(x) = infinity

This means that as x gets arbitrarily close to 0 (but is never equal to 0), the value of f(x) gets arbitrarily large (in absolute value).

Limits are also used to describe the behavior of sequences and series. For example, the series 1 + 1/2 + 1/4 + 1/8 + … is an infinite sum of numbers that get smaller and smaller. The limit of this series as the number of terms approaches infinity is 2, which means that the sum of the series gets arbitrarily close to 2 as we add more and more terms.

Properties of Limits

The properties of limits are important concepts in calculus that help evaluate the limits of functions and prove theorems. There are several fundamental properties of limits that you should be aware of:

Sum/Difference Rule:

The limit of the sum or difference of two functions is equal to the sum or difference of their limits. In other words, if the limit of f(x) as x approaches a is L, and the limit of g(x) as x approaches a is M, then the limit of [f(x) ± g(x)] as x approaches a is L ± M.

Limx→c [f(x) ± g(x)] = Limx→c [f(x)] ± Limx→c [g(x)] = L ± M

1.    Product Rule:

The limit of the product of two functions is equal to the product of their limits. If the limit of f(x) as x approaches a is L, and the limit of g(x) as x approaches a is M, then the limit of [f(x) × g(x)] as x approaches a is L × M.

Limx→c [f(x) x g(x)] = Limx→c [f(x)] x Limx→c [g(x)] = L x M

2.    Quotient Rule:

The limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the denominator is not zero. That is if the limit of f(x) as x approaches a is L, and the limit of g(x) as x approaches a is M (where M is not equal to 0), then the limit of [f(x) / g(x)] as x approaches a is L / M.

Limx→c [f(x) / g(x)] = Limx→c [f(x)] / Limx→c [g(x)] = L / M

3.    Power Rule:

The limit of a function raised to a power is equal to the limit of the function raised to that power. If the limit of f(x) as x approaches a is L and n is a positive integer, then the limit of [f(x) ^ n] as x approaches a is L ^ n.

Limx→c [f(x)]n = [Limx→c f(x)]n = Ln

4.    Constant Multiple Rule:

The limit of constant times a function is equal to the constant times the limit of the function. That is if c is a constant and the limit of f(x) as x approaches a is L, then the limit of [c × f(x)] as x approaches a is c × L.

Limx→c [c*f(x)] = c* Limx→c [f(x)]

5.    Limit of a Composite Function:

The limit of a composite function is equal to the limit of the outer function evaluated at the limit of the inner function. That is, if the limit of g(x) as  approaches  is  and the limit of f(x) as x approaches a isM, then the limit of [g(f(x))] as x approaches a is g(M).

Limx→c [f(x)] = M and Limx→c [g(x)] = L

Then

Limx→c g[f(x)] = Limx→c g[M]

Important Formulas

  • Lim(x→a) (x^n-a^n)/(x-a) = na^(n-1)  for al ral values of n.
  • Lim(θ→0) sinθ/θ = 1
  • Lim(θ→0) tanθ/θ = 1
  • Lim(θ→0) (1-cosθ)/θ = 0
  • Lim(x→0) e^x = 1
  • Lim(θ→0) cosθ = 0
  • Lim(x→0)(e^x-1)/x = 1
  • Lim(x→∞)(1+1/x)^x = e

Examples of Limit

Example:

Find the limit of the function f(x)=(x^2- 1)/(x – 1) as x approaches 1.

Solution:

Direct substitution of x = 1 gives an indeterminate form of 0/0, so we need to simplify the expression first. We can factor the numerator as (x + 1) (x – 1) and cancel the common factor of (x – 1) in the numerator and denominator, giving:

f(x) = x + 1

Now, we can substitute x = 1 directly to obtain the limit:

limx→1 f(x) = limx→1 (x + 1) = 2

Therefore, the limit of f(x) as x approaches 1 is 2.

A limit calculator with steps can be used to get the step-by-step solution of the complex limit problems to reduce the difficulty of calculating them manually.

Example

Find the limit of the function f(x) = ((cos(x)- 1))/x^2  as x approaches 0.

Solution:

Lim(x→0) (cos(x)-1)/x^2   

Applying L’Hospital Rule

=Lim(x→0) sin(x)/2x

Applying L’Hospital Rule

=Lim(x→0)-cos(x)/2

Put x=0

=cos(0)/2

Simplify

-cos(0)/2 = -1/2

Hence,

Lim(x→0) (cos(x)-1)/x^2 = -1/2

Conclusion

To sum up, limits are a crucial concept in mathematics that describe the behavior of functions, sequences, and series as their inputs or indices approach certain values.

They are used extensively in calculus to define derivatives and integrals and are a key tool for modeling and analyzing complex systems in various areas of science and engineering. Developing an understanding of the concept of a limit is essential for mastering calculus and many other areas of mathematics and science.

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