Limit Calculator: The best tool to find the limit of a function

Limit Calculator


Lim( )

Use inf for +∞ and -inf for -∞

Side:






Solution:

If you are currently studying how to evaluate limits of functions, without a doubt the Limit Calculator that we put at your disposal here will be of great help to you. With the Limits Calculator you can calculate One-Sided Limits and Two-Sided Limits.

Instructions for using the Limit Calculator

Looking at the calculator you will have noticed that it is very intuitive, which makes its use very simple. To find the limit of a function you just have to follow these simple steps:

Instructions for using the limit calculator
  1. Write the function whose limit you want to evaluate, for this you must make use of the list of valid functions.
  2. Then choose the variable and what value the variable approaches.
  3. Select the type of limit you want to calculate, you have three options:
    • Two-Sided Limits
    • One-Sided Limits:
      • Right-Hand (+)
      • Left-Hand (-)
  4. Finally press the calculate button to get the result.
Valid functions and symbols Description
sqrt() Square root
ln() Natural logarithm
log() Logarithm base 10
^ Exponents
abs() Absolute value
sin(), cos(), tan(), csc(), sec(), cot() Basic trigonometric functions
asin(), acos(), atan(), acsc(), asec(), acot() Inverse trigonometric functions
sinh(), cosh(), tanh(), csch(), sech(), coth() Hyperbolic functions
asinh(), acosh(), atanh(), acsch(), asech(), acoth() Inverse hyperbolic functions
pi PI number (π = 3.14159...)
e Neper number (e= 2.71828...)
i To indicate the imaginary component of a complex number.
inf

Definition of limit of a function

The limit of a function can be defined as the value L that f(x) appears to approach when the independent variable x approaches a certain value x0. The formal definition based on what was said above is as follows:

 lim xx0 f(x) = L

For example, consider the function f(x) = x^2. The limit of this function as x approaches 2, denoted as lim(x→2)f(x), is 4. This is because as x gets closer and closer to 2, the value of f(x) gets closer and closer to 4.

The limit of a function can be used to describe the behavior of the function near a certain point, even if the function is not defined at that point. For example, the function g(x) = 1/x is not defined at x = 0, but we can still talk about the limit of this function as x approaches 0. In this case, the limit is infinity, because as x gets closer and closer to 0, the value of g(x) gets larger and larger.

Limits are an important concept in mathematics because they allow us to describe the behavior of functions at points where the function may not be defined, and they play a central role in the development of calculus.

Limit Rules

The properties of limits are the set of algebraic rules and procedures used to calculate them. The concept of a limit is fundamental to calculus and finding its value does not have to be a complicated task, as long as you know these rules. Below is a list of the main properties of limits:
Limit of a constant  lim xa k = k
Constant Multiple Rule  lim xa (k·f(x)) = k· lim xa (f(x))
Sum of functions  lim xa (f(x)+g(x)) =  lim xa (f(x))+ lim xa (g(x))
Difference of functions  lim xa (f(x)-g(x)) =  lim xa (f(x))- lim xa (g(x))
Product of functions  lim xa (f(xg(x)) =  lim xa (f(x))· lim xa (g(x))
Quotient of functions  lim xa f(x)g(x) =  lim xa (f(x)) lim xa (g(x))&nbsp ; &nbspif&nbsp&nbsp&nbsp lim xa (g(x)) ≠ 0
Limit Exponent rule  lim xa (f(x)n) = ( lim xa (f(x)))n
Root law limits  lim xa ( n  f(x)) =  n   lim xa (f(x))
Limit of polynomial function  lim xa (p(x)) = (p(a))

Methods for finding the limit of a function

  1. Direct substitution: This method involves simply plugging in the value that x is approaching into the function and seeing what the output is. If the output is a finite number, then that is the value of the limit. If the output is infinity or an undefined value, then we need to use a different method.

  2. Factoring: This method involves factoring the expression for the function into simpler terms and then using the properties of limits to find the limit. 

  3. L’Hopital’s rule: This method involves taking the derivative of the function with respect to x and finding the limit of the derivative. If the derivative has a finite limit as x approaches a certain value, then the original function also has a finite limit at that value.

  4. Series expansion: This method involves expanding the function in a Taylor series around the point where the limit is being taken. The limit can then be calculated by taking the appropriate term in the expansion.

  5. Asymptotic expansion: This method involves finding an expression for the function that becomes exact in the limit as x approaches a certain value. The limit can then be calculated by taking the appropriate term in the expansion.

  6. Numerical methods: In some cases, it may be necessary to use numerical methods to approximate the limit of a function. One common method is to use a computer program to evaluate the function at a large number of points near the point where the limit is being taken and then use those values to estimate the limit.

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