In calculus, a limit is a fundamental concept that describes the behavior of a function as its argument approaches a specific value. The limit of a function helps to understand how a function behaves near a certain point, even if it does not attain that point explicitly. The concept of limits is essential for defining key calculus concepts like derivatives, continuity, and integrals.
About Limit Formula
The derivative of a function is calculated using the limit formula. As the input approaches the specified value, the function's value approaches the limit. Limits are used to bring approximations used in calculations as near to the true value of the quantity as possible.
What do you mean by Limit Formula?
The limit formula is a representation of a function's behaviour at a certain point, and it examines that function. Limit specifies how a number that is reliant on an independent variable behaves when that independent variable approaches or approaches a specific value.
Limit Formula
As a function of x, let y = f(x). If f(x) adopts indeterminate form at x = a, we can analyse the values of the function that are very close to a. If these values tend to a definite unique number as x approaches a, that number is known as the limit of f(x) at x = a. The formula can be written as:
Here, f(x) is a mathematical function and x is a variable that is getting close to having a value. When x approaches a, it is viewed as the limit of a function of x equals A. Formulas of Limits
A few limitations formulas are shown in the graphic below.
Limit Formula in Calculus
Introduction to Limits:
In calculus, a limit is a fundamental concept that describes the behavior of a function as its argument approaches a specific value. The limit of a function helps to understand how a function behaves near a certain point, even if it does not attain that point explicitly. The concept of limits is essential for defining key calculus concepts like derivatives, continuity, and integrals.
Limit Formula:
The limit of a function f(x)f(x) as xx approaches a number cc is expressed as:
limx→cf(x)=L\lim_{x \to c} f(x) = L
Where:
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cc is the point that xx is approaching.
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LL is the value that f(x)f(x) approaches as xx gets arbitrarily close to cc.
In simple terms, this means that as xx approaches the value cc, the function f(x)f(x) gets closer to the value LL.
Basic Limit Formulas:
There are several standard limit formulas that help simplify the process of evaluating limits. Below are some of the basic limit formulas commonly used in calculus:
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Limit of a Constant Function:
limx→ck=k\lim_{x \to c} k = kFor any constant kk, the limit of kk as xx approaches any value cc is simply kk.
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Limit of xnx^n (Power Functions):
limx→cxn=cn\lim_{x \to c} x^n = c^nThe limit of xnx^n as xx approaches cc is simply cnc^n, where nn is a positive integer.
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Limit of Rational Functions (Quotients):
limx→cf(x)g(x)=limx→cf(x)limx→cg(x)iflimx→cg(x)≠0\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \quad \text{if} \quad \lim_{x \to c} g(x) \neq 0The limit of a quotient is the quotient of the limits, provided the denominator does not approach zero.
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Limit of exe^x (Exponential Functions):
limx→cex=ec\lim_{x \to c} e^x = e^cThe limit of the exponential function exe^x as xx approaches cc is ece^c.
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Limit of ln(x)\ln(x) (Natural Logarithmic Functions):
limx→cln(x)=ln(c)\lim_{x \to c} \ln(x) = \ln(c)The natural logarithm of xx has the limit ln(c)\ln(c) as xx approaches cc.
Common Limits:
Some common limits that students often use are:
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Limit of sin(x)\sin(x) as xx approaches 0:
limx→0sin(x)=0\lim_{x \to 0} \sin(x) = 0 -
Limit of cos(x)\cos(x) as xx approaches 0:
limx→0cos(x)=1\lim_{x \to 0} \cos(x) = 1 -
Limit of sin(x)x\frac{\sin(x)}{x} as xx approaches 0:
limx→0sin(x)x=1\lim_{x \to 0} \frac{\sin(x)}{x} = 1 -
Limit of ex−1x\frac{e^x - 1}{x} as xx approaches 0:
limx→0ex−1x=1\lim_{x \to 0} \frac{e^x - 1}{x} = 1
Indeterminate Forms and L'Hopital's Rule:
In some cases, limits result in forms like 00\frac{0}{0}, ∞−∞\infty - \infty, or ∞∞\frac{\infty}{\infty}, which are called indeterminate forms. These cases require special techniques to evaluate the limits.
L'Hopital's Rule is a method for resolving indeterminate forms. It states:
limx→cf(x)g(x)=limx→cf′(x)g′(x)\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
provided that the limit on the right side exists, and both f(x)f(x) and g(x)g(x) are differentiable.
One-Sided Limits:
A one-sided limit refers to the value that a function approaches as xx approaches a particular value from either the left or the right.
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Left-hand limit:
limx→c−f(x)\lim_{x \to c^-} f(x)This refers to the limit of f(x)f(x) as xx approaches cc from the left (i.e., xx is smaller than cc).
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Right-hand limit:
limx→c+f(x)\lim_{x \to c^+} f(x)This refers to the limit of f(x)f(x) as xx approaches cc from the right (i.e., xx is greater than cc).
If the left-hand and right-hand limits are equal, the two-sided limit exists.
Download free pdf of Limit Formula Its Use And Solved Examples
Frequently Asked Questions
To find the limit of a function, follow these steps:
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Substitute the value of xxx directly into the function (if possible). This works when the function is continuous at the point.
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If direct substitution leads to an indeterminate form like 0/00/00/0, try factoring, simplifying, or using L'Hopital’s Rule.
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If needed, use limit laws like the sum, product, or quotient rules to break down complex expressions into simpler terms.
The Limit Formula is a foundational concept in calculus because it forms the basis for derivatives and integrals. The derivative of a function is defined as the limit of the average rate of change as the interval approaches zero, and the integral uses limits to calculate the area under curves. Limits help us understand the behavior of functions at specific points, even if they are undefined at those points.
Yes, limits can be infinite. For example, if the function increases or decreases without bound as xxx approaches a particular value, the limit is said to be infinite. It is written as:
limx→af(x)=∞orlimx→af(x)=−∞\lim_{{x \to a}} f(x) = \infty \quad \text{or} \quad \lim_{{x \to a}} f(x) = -\inftyx→alimf(x)=∞orx→alimf(x)=−∞
This typically happens when a function has a vertical asymptote at the given point.
A logarithm is the inverse operation of exponentiation. It answers the question: to what power must a specific base be raised, to get a particular number? For example, logb(x)=y\log_b(x) = ylogb(x)=y means that by=xb^y = xby=x, where bbb is the base, xxx is the result, and yyy is the exponent.
To solve logarithmic equations, you can:
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Use logarithmic properties to simplify the equation.
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Convert the logarithmic equation into an exponential equation.
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Solve for the unknown variable using algebraic techniques.
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