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Calculus 1 – Introduction to Limits
Calculus 1 – Introduction to Limits

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Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function. Let us discuss the definition and representation of limits of the function, with properties and examples in detail.

1. What Are Limits?
2. Limits and Functions
3. Properties of Limits
4. Limit of a Function of Two Variables
5. Limits of Complex Functions
6. Limits of Exponential Functions
9. FAQs on Limits

What Are Limits?

Limits in maths are unique real numbers. Let us consider a real-valued function “f” and the real number “c”, the limit is normally defined as \(\lim _{x \rightarrow c} f(x)=L\). It is read as “the limit of f of x, as x approaches c equals L”. The “lim” shows the limit, and fact that function f(x) approaches the limit L as x approaches c is described by the right arrow.

Limits and Functions

A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and left-hand limits exist.

  • When the \(\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\mathrm{A}^{+}\) given the values of f near x to the right of a. This value is said to be the right hand limit of f(x) at a.
  • When the \(\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\mathrm{A}^{-}\) given the values of f near x to the left of a. This value is called the left hand limit of f(x) at a.
  • The limit of a function exists if and only if the left-hand limit is equal to the right-hand limit. \(\lim _{x \rightarrow a^{-1}} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a^{+}} \mathrm{f}(\mathrm{x})=\mathrm{L}\)

Note: The limit of the function exists between any two consecutive integers.

Properties of Limits

Here are some properties of the limits of the function: If limits \( \lim _{x \rightarrow a}\) f(x) and \(\lim _{x \rightarrow a}\) g(x) exists, and n is an integer, then,

  • Law of Addition: \(\lim _{x \rightarrow a}[f(x)+g(x)]=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)\)
  • Law of Subtraction: \( \lim _{x \rightarrow a}[f(x)-g(x)]=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)\)
  • Law of Multiplication: \(\lim _{x \rightarrow a}[f(x) \cdot g(x)]=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)\)
  • Law of Division: \( \lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}, \text { where } \lim _{x \rightarrow a} g(x) \neq 0\)
  • Law of Power: \(\lim _{x \rightarrow a} c=c \)

Special Rules:

1. \(\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{(n-1)}\), for all real values of n.
2. \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\)
3. \(\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}=1\)
4. \(\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}=0\)
5. \(\lim _{\theta \rightarrow 0} \cos \theta=1\)
6. \(\lim _{x \rightarrow 0} e^{x}=1\)
7. \(\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1\)
8. \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e\)

Limit of a Function of Two Variables

If we have a function f(x, y) which depends on two variables x and y, then this given function has the limit, say, C as (x,y) → (a,b) provided that ϵ > 0, there exists Δ > 0 such that |f(x, y)-C| < ϵ whenever 0 < \(\sqrt{(x-a)^{2}+(y-b)^{2}}\) < Δ . It defined as \(\lim _{(x, y) \rightarrow(a, b)}\) f(x,y) = C.

Limits of Functions and Continuity

Limits of the function and continuity of the function are closely related to each other. Functions can be continuous or discontinuous. For a function to be continuous, if there are small changes in the input of the function then must be small changes in the output.

In elementary calculus, the condition f(X) →λ as x → a means that the number f(x) can be made to lie as close as we like to the number λ as long we take the number unequal to the number a but close enough to a. Which shows that f(a) might be very far from λ and there is no need for f(a) even to be defined. The very important result we use for the derivation of function is: f'(a) of a given function f at a number a can be thought of as,
f'(a) =\(\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\)

Limits of Complex Functions

To differentiate functions of a complex variable follow the below formula:
The function \(f(z)\) is said to be differentiable at \(z=z_{0}\) if
\(\lim _{\Delta z \rightarrow 0} \frac{f\left(z_{0}+\Delta z\right)-f\left(z_{0}\right)}{\Delta z}\) exists. Here \(\Delta \mathrm{z}=\Delta \mathrm{x}+\mathrm{i} \Delta \mathrm{y}\)

Limits of Exponential Functions

For any real number x, the exponential function f with the base a is f(x) = ax where a >0 and a not equal to zero. Below are some of the important laws of limits used while dealing with limits of exponential functions.
For f(b) >1

  • \(\lim _{x \rightarrow \infty} b^{x}=\infty\)
  • \(\lim _{x \rightarrow-\infty} b^{x}=0\)

For \(0<b<1\)

  • \(\lim _{x \rightarrow \infty} b^{x}=0\)
  • \(\lim _{x \rightarrow-\infty} b^{x}=\infty\)

☛ Also Check:

Examples of Limits

  1. Example 1: Check for the limit, \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\)


    Since we have modulus function in the numerator, so let us evaluate right hand and left-hand limits first.

    RHL=\(\lim _{h \rightarrow 0+} \frac{|\sin (h)|}{h}=1\)

    LHL=\(\lim _{h \rightarrow 0^{-}} \frac{|\sin (-h)|}{-h}=-1\)

    Answer: As RHL and LHL revert different values, therefore, the given limit does not exist.

  2. Example 2: Find the limit of the function. lim_(x→0) (tan x)/(sin x)

    Solution: lim\(_{x→0}\)(tan x)/(sin x)

    We know, tan x = sin x / cos x

    = lim\(_{x→0}\) (1/cos x)

    = 1

    Answer: The limit lim\(_{x→0}\) (tan x)/(sin x) =1

  3. Example 3: Evaluate lim\(_{x→0}\) sin(2x)


    lim\(_{x→0}\) sin(2x) = lim\(_{x→0}\) [sin(2x)/2×(2)]

    = 2 lim\(_{x→0}\) sin(2x)/2

    = 2

    Answer: The limit lim\(_{x→0}\) sin(2x) = 2

Practice Questions on Limits

FAQs on Limits

What is the Limit Formula?

Limits formula:- Let y = f(x) as a function of x. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained a unique number is called the limit of f(x) at x = a.

What Are Limits in Calculus?

A limit tells us the value that a function approaches as that function’s inputs get closer and closer(approaches) to some number. The idea of a limit is the basis of all differentials and integrals in calculus.

When Can a Limit Not Exist?

A common situation where the limit of a function does not exist is when the one-sided limits exist and are not equal: the function “jumps” at the point. The limit at x→0 does not exist.

Why Do We Use Limits in Mathematics?

Limit, a mathematical concept based on the idea of closeness, is used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.

How Do You Know if a Limit Is One-Sided?

A one-sided limit is a value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn’t defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1.

How Are Calculus Limits Used in Real Life?

Limits are also used as real-life approximations to calculating derivatives. So, to make calculations, engineers will approximate a function using small differences in the function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals.

What Is the Limit of a Sine Function?

Since sin(x) is always somewhere in the range of -1 and 1, we can set g(x) equal to -1/x and h(x) equal to 1/x. We know that the limit of both -1/x and 1/x as x approaches either positive or negative infinity is zero, therefore the limit of sin(x)/x as x approaches either positive or negative infinity is zero.

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