Is Surface Continuous if Limit Can Be Evaluated

This article covers an important topic of calculus that is limits and continuity, calculus is a part of mathematics that essentially deals with the study of change in the value of a function for the changes in the points of the domain.
What is a limit? Limit in general is defined as:
\(x \longrightarrow a\), and \(f(x) \longrightarrow l\) here l is called the limit to the function f(x), which can be written as \(\lim _{x \rightarrow a} f(x)=l\)

Similarly, a function is said to be continuous if the left-hand limit, right-hand limit, and the value of the function say at a point x = c exist and are equal to each other, then the function f is declared to be continuous at the point x = c. You will get a clear idea about the left/right-hand limit in the upcoming paragraphs of the article.

\(\lim_{x\to c}=f\left(x\right)=f\left(c\right)\)

Discover major theories and properties of limits and continuity, comprising limits and continuity formulas, discontinuity condition, existence, and more such concepts about Limits and Continuity.

What is a Limit?

Consider \(y=f(x)\) be a function. Then for a function, we say \(\lim _{x \rightarrow a} f(x)\) exists i.e, \(\lim _{x \rightarrow a} f(x)=l\) where l is a finite value.

Similarly \(\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=l=\lim _{x \rightarrow a} f(x)\)
Here \(\lim _{x \rightarrow a^{+}} f(x) {\text {is named the right-hand limit and is denoted by } \mathrm{RHL} \text { and }} \lim _{x \rightarrow a^{-}} f(x)\) is called the left-hand limit and is expressed by LHL.

This implies that there are two modes x could approach a number a either from the left or from the right, i.e., all the values of x near a could be less than a or greater than a. This generally leads to two limits – the right-hand limit and the left-hand limit. The right-hand limit of a function f(x) is those values of f(x) which are composed of the values of f(x) when x tends to a from the right. Similarly, for the left-hand limit, the condition follows.

To find RHL, we can substitute x = a + h and replace x → a+ by h → 0 and then simplify

\(\lim _{h \rightarrow 0} f(a+h)\)
i.e. \(\lim _{x \rightarrow a^{+}} f(x)=\lim _{h \rightarrow 0} f(a+h)\)

Similarly, to find LHL, we can put x = a – h and replace x → a- by h → 0 and then simplify

\(\lim _{h \rightarrow 0} f(a-h)\)
i.e \(\lim _{x \rightarrow a^{-}} f(x)=\lim _{h \rightarrow 0} f(a-h)\)

If you are reading about Limits And Continuity you should also read about the Evaluation of Limits here!

One-Sided Limit

A two-sided limit \(\lim _{x \rightarrow a} f(x)\) uses the values of x into a statement that are both greater than and less than a. Where's a one-sided limit from the left \(\lim _{x \rightarrow a-} f(x)\) or from the right \(\lim _{x \rightarrow a+} f(x)\) takes exclusive values of x smaller/greater than a respectively.

Difference Between Limit and Continuity

The common difference between Limit and Continuity is listed below:

Limit	Continuity
A limit can be defined as a number approached by the function when an independent function's variable comes to a particular value.	The straightforward approach to test for the continuity of a function is to analyse whether a pen/pencil can outline the graph of a function without raising the pen/pencil from the paper.
If\(x\ \longrightarrow a \text{ and } f\left(x\right)\longrightarrow\ l\) then 'l' is said to be the limit to the function f(x). which can be mathematically written as: \(\lim_{x\to a} f\left(x\right)=\ l\)	Think if f is a real function on a subset of the real numerals and c is a point in the domain of f. Then the function f is continuous at c if: \(\lim_{x\to c}f\left(x\right)=\ f\left(c\right)\)

If you have mastered Limits and Continuity, you can also learn about Integral Calculus here.

What is Continuity?

Continuity Meaning: Consider any function f(x) which is defined for x = a is stated to be continuous at x = a, if:
f(a) is a finite value.
The limit of the function f(x) as x → a exists and is equal to the value of f(x) at x = a. i.e.
\(\lim_{x\to a}f\left(x\right)=l=f\left(a\right)\)

Thus f(x) is continuous at x=a if we have f(a+0)=f(a-0)=f(a), otherwise the function is discontinuous at x=a.

Learn more about Differential Calculus with this article.

Discontinuity of a Function

A function f(x) which is not continuous at a point x = a, then a function f(x) is said to be discontinuous at x = a.

Types of Discontinuity

Check the types of discontinuity in the details below:

Infinite Discontinuity : Infinite discontinuity is defined as a branch of discontinuity wherever a vertical asymptote is present at x = a, and f(a) is not defined. This is also termed Asymptotic Discontinuity.
Jump Discontinuity : Jump discontinuity is said to occur when; \(\lim _{x \rightarrow a+} f(x) \neq \lim _{x \rightarrow a-} f(x)\) however, both the limits are finite. \(\lim_{x\to a+}f\left(x\right)\ne\lim_{x\to a-}f\left(x\right)\)
Positive Discontinuity : Positive discontinuity occurs when a function has a predefined two-sided limit at x = a, but either f(x) is not defined at a, or its value is not identical to the limit at a.

Limits and Continuity Formulas

Some common and useful limits and continuity formulas are discussed below:

Limit Formulas

Starting with the limit formulas; which covers trigonometric, logarithmic, exponential followed by the algebra of limits, L' Hospital's rule, sandwich theorem and more.

Some Important Limit Formulas

\(\lim_{x\to a}\left[\frac{x^n-a^n}{x-a}\right]=na^{n-1}\text{ for any positive integer n.}\)

\(\lim_{x\to0}\left[\frac{\sin x}{x}\right]=1\)

\(\lim_{x\to0}\left[\frac{\tan\ x}{x}\right]=1\)

\(\lim_{x\to0}\sin x=0\)

\(\lim_{x\to0}\ \cos x=1\)

\(\lim_{x\to0}\left[\frac{a^x-1}{x}\right]=\ln\ a,\ a\ >0\)

\(\lim_{x\to0}\left[\frac{e^x-1}{x}\right]=1\)

\(\lim_{x\to0}\left[1+x\right]^{\frac{1}{x}}=e\)

\(\lim_{x\to\infty}\left[1+\frac{a}{x}\right]^x=e^a\)

\(\lim_{x\to\infty}\left[1+\frac{1}{x}\right]^x=e\)

\(\lim_{x\to0}\left[\frac{\log_a\left(1+x\right)}{x^m}\right]=\log_ae,\ \left(a>0\ and\ a\ \ne1\right)\)

\(\lim_{x\to0}\left[\frac{\log\left(1+x\right)}{x}\right]=1\)

Algebra of Limits

Some of the algebraic formulas regardings limits are as follows:

Algebra of Limits

\(\lim_{x→a}k=k\text{ here k is constant}\)

\(\lim_{x→a}kf\left(x\right)=k\lim_{x→a}f\left(x\right)\)

\(\lim_{x→a}\left[f(x)\pm g(x)\right]=\lim_{x→a}f(x)\pm\lim_{x→a}g(x)\)

\(\lim_{x→a}\left[f(x).g(x)\right]=\lim_{x→a}f(x)\ .\ \lim_{x→a}g(x)\)

\(\lim_{x→a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}\)

Sandwich Theorem

Consider f, g, and h be functions such that f(x) ≤ g(x) ≤ h(x) for all x in some neighborhood of the point a(except possibly at x = a) and if , then:

\(\lim_{x\to a}f\left(x\right)=l=\lim_{x\to a}h\left(x\right),\ then\ \lim_{x\to a}g\left(x\right)=l\)

Continuity Formulas

If f(a) is defined and \(\lim_{x→a}f(x)\) exists; \(\lim_{x→a^+}f(x)=\lim_{x→a^−}f(x)=f(a)\)

Then a function f is stated to be continuous at the point x = a.

Algebra of continuous functions

In the earlier heading, we learnt some algebra of limits, so now let us learn some algebra of continuous functions. If we are given two real functions f and g that are continuous at a real number d. Then the formula below holds good:

f + g is continuous at x = d.
f – g is continuous at x = d.
f . g is continuous at x = d.
\(\frac{f}{g}\) is continuous at x = d, (when g (d) ≠ 0).

Key Takeaways of Limits and Continuity

For a function \(f(x)\) the limit of the function at a point \(x=a\) is the value the function attains at a point that is very near to \(x=a\).
If the limit is defined in terms of a number that is smaller than a: then the limit is called the left-hand limit. It is denoted as \( x \rightarrow a^{-} \)
If the limit is defined in terms of a number that is greater than a then: the limit is called the right-hand limit. It is denoted as \( x \rightarrow a^{+} \)
The limit of a function \( f(x) at x=a \) exists: only when its left-hand limit and right-hand limit exist and are equal and possess a finite value.
i.e. \( \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x) \)
A function is said to be continuous over a range: if it′s graph is a single unbroken curve.
If a function \(f(x)\) is continuous at \(x=x_{\circ}\)
i.e. \( \lim_{x\to x_{\circ}^+}f(x)=\lim_{x\to x_{\circ}^-}f(x)=\lim_{x\to x_{\circ}}f(x) \)

Limits and Continuity Problems with Solutions

With all the knowledge of limits followed by continuity including definition, formulas, types and key takeaways it's time to practice some examples for more clarity:

Solved Example 1: Find the limits for the expression given by \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\)

Solution:

Given function \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\) substitute the limits.

Therefore \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\) =1

Solved Example 2: Evaluating the limit for the function;\(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x^2-4\right)}\).

Solution: Given function is \(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x^2-4\right)}\).

Open the function into its component as shown:

\(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}\)

Now substituting the limits we get:

\(\frac{x\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}=\frac{x\left(x-2\right)}{\left(x+2\right)}=\frac{1\left(1-2\right)}{\left(1+2\right)}=\frac{-1}{3}\)

Solved Example 3: Check for the continuity of the function f given by the expression f (x) = 4x + 5 at x = 1.

Solution: First check for the limit:

\(\lim_{x\to1}\ f\left(x\right)=\lim_{x\to1}\ \left(4x+5\right)=9\)

Now check for f(1)

f (x) = 4x + 5

f (1) = 4(1) + 5=9

Hence \(\lim_{x\to1}\ f\left(x\right)=9=f\left(1\right)\)

Therefore, the given function is continuous at x = 1.

Through this article, we learned about limits and continuity a part of mathematical calculus and explored concepts of limits via existence, properties, indeterminate form, Sandwich Theorem followed by Exponential, Logarithmic, and Trigonometric limits in succession with continuity by covering topics like the continuity of a function in the interval, properties, and types of discontinuity along with key ideas of Infinitesimals.

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Limits and Continuity FAQs

Q.1 What is continuity in maths?

Ans.1 A function is said to be continuous if the left-hand limit, right-hand limit, and the value of the function at a point x = c exist and are equal to each other.

Q.2 When a function is said to be discontinuous?

Ans.2 A function f(x) which is not continuous at a point x = a, then function f(x) is said to be discontinuous at x = a.

Q.3 What is the statement for Sandwich Theorem?

Ans.3 Consider f, g, and h be functions such that f(x) ≤ g(x) ≤ h(x) for all x in some neighborhood of the point a(except possibly at x = a) and if , then \(\lim_{x\to a}f\left(x\right)=l=\lim_{x\to a}h\left(x\right),\ then\ \lim_{x\to a}g\left(x\right)=l\)

Q.4 State the condition for continuity in the open interval?

Ans.4 A function f(x) is stated to be continuous over an open interval (a, b) if it is continuous at each point over the interval (a, b).

Q.5 What is the condition for continuity in the close interval?

Ans.5 A function f(x) is declared to be continuous over a closed interval [a, b] if it is continuous over the open interval (a, b) and is continuous at point a from the right and continuous at point b from the left.

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