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.

Steps:

• 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.

Steps:

• 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.

Steps:

• 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, \(lim_{x→0}​\frac{sin(x)​}{x}=1\).

Steps:

• 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.

Steps:

• 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.

Steps:

• 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.

Steps:

• 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:

\(lim_{x→c}​f(x)=f(c)\)

2. Rational Function:

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

Limit Calculation:

\(lim_{x→c}​f(x)=\frac{P(c)}{Q(c)}\)

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.