How to Calculate Limits of Functions

The concept of a limit is foundational in calculus. It represents the value a function approaches as its input (or variable) approaches a certain point. Calculating these limits gives us a glimpse into the behavior of functions near specific points, even if they are undefined at those very points. Let's walk through the steps of calculating these elusive values.

How to Calculate Limits of Functions

Step-by-step Guide to Calculate Limits of Functions

Here is a step-by-step guide to calculating the limits of functions:

Step 1: Direct Substitution

This is the simplest method. If a function is continuous at the point of interest, you can directly substitute that point into the function.


  • a. Plug the point of interest into the function.
  • b. Evaluate the function.
  • c. If you obtain a real number, that’s your limit.

Step 2: Factoring and Simplifying

For functions that become indeterminate (like \(\frac {0}{0}\)) when you directly substitute, try simplifying the function.


  • a. Factor the function if possible.
  • b. Cancel out common terms in the numerator and denominator.
  • c. Try direct substitution again.

Step 3: Rationalizing

For functions with radicals where direct substitution leads to an indeterminate form, use rationalization.


  • a. Multiply the numerator and denominator by the conjugate of the radical expression.
  • b. Simplify the resulting expression.
  • c. Retry direct substitution.

Step 4: Using Special Limit Properties

Certain functions have well-known limit properties. For instance, \(lim_{x→0}​\frac{sin(x)​}{x}=1\).


  • a. Recognize the form of the function.
  • b. Apply the known limit property.

Step 5: L’Hôpital’s Rule

For indeterminate forms like \(\frac{0}{0}\) or \(\frac{∞}{∞}\), L’Hôpital’s rule is a powerful tool. It states that the limit of the ratio of two functions can be found by taking the limit of the ratio of their derivatives.


  • a. Differentiate the numerator.
  • b. Differentiate the denominator.
  • c. Retry direct substitution.
  • d. If you still get an indeterminate form, repeat the differentiation process until you obtain a determinate form.

Step 6: Squeeze (or Sandwich) Theorem

For functions trapped between two others that approach the same limit, you can conclude that the function in question has the same limit as well.


  • a. Identify two other functions that ‘squeeze’ or ‘sandwich’ the given function.
  • b. Calculate the limits of these squeezing functions.
  • c. If both functions approach the same limit, your function has that limit as well.

Step 7: Using Graphs

Sometimes, visualizing the function can provide insight into its limit.


  • a. Plot the function using graph paper or graphing software.
  • b. Observe the behavior of the function as it approaches the point of interest.
  • c. Make an educated estimate of the limit based on the graph.

Calculation of the Limits of Various Functions

1. Polynomial Function:

Definition: Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients

Limit Calculation:

Calculating the limit for a polynomial function is straightforward. Due to their continuous nature everywhere, you can directly substitute the value into the function:


2. Rational Function:

Definition: Rational functions are the ratio of two polynomial functions.

Limit Calculation:


3. Root Function:

Definition: Root functions involve radicals, like the square root, cube root, etc.

Limit Calculation: If \(\sqrt[n]{f(x)}\), where \(f\) is a function with domain \(D_f\),then:

\(lim_{x→c}\sqrt[n]{f(x)}=\sqrt[n]{f(c)}​; c∈D_f\) and \(f(c)≥0\)

4. Exponential Function:

Definition: Exponential functions have a variable in the exponent of a fixed base. The most common base is the natural number \(e\).

Limit Calculation: \(f(x)=a^x\)

  • \(lim_{x→+∞}a^x =+∞\) (If \(a>1\))
  • \(lim_{x→-∞}a^x =0\) (If \(a>1\))
  • \(lim_{x→+∞}a^x =0\) (If \(0<a<1\))
  • \(lim_{x→-∞}a^x =+∞\) (If \(0<a<1\))

Final Word:

Calculating limits requires a combination of techniques, analytical thinking, and sometimes a touch of intuition. While some limits can be determined quickly, others might require more in-depth exploration. With practice and persistence, you’ll find that discerning the behavior of functions at specific points becomes a rewarding mathematical endeavor.

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