Calculus Limits: Factoring Example

Calculus is a branch of mathematics that deals with continuous change. One of the fundamental concepts in calculus is the limit. A limit is the value that a function approaches as the input approaches some value. Limits are essential for understanding continuity, derivatives, and integrals.

One common technique for evaluating limits is factoring. Factoring allows us to simplify the expression and eliminate any factors that would make the denominator zero. This is particularly useful when dealing with limits that result in indeterminate forms, such as 0/0 or ∞/∞.

Example

Let’s consider the following limit:

lim_x→2 (x² – 4) / (x – 2)

If we directly substitute x = 2 into the expression, we get 0/0, which is an indeterminate form. To evaluate this limit, we can factor the numerator:

(x² – 4) = (x – 2)(x + 2)

Now, we can rewrite the limit as:

lim_x→2 [(x – 2)(x + 2)] / (x – 2)

Since x ≠ 2, we can cancel out the (x – 2) factor in the numerator and denominator:

lim_x→2 (x + 2)

Now, we can directly substitute x = 2 into the expression:

lim_x→2 (x + 2) = 2 + 2 = 4

Therefore, the limit of the function as x approaches 2 is 4.

Steps to Solve Limits by Factoring

1. Substitute the value that x is approaching into the function.
2. If the result is an indeterminate form (0/0 or ∞/∞), factor the numerator and denominator.
3. Cancel out any common factors.
4. Substitute the value that x is approaching into the simplified expression.

Conclusion

Factoring is a powerful technique for evaluating limits in calculus. By simplifying the expression and eliminating any factors that would make the denominator zero, we can often obtain a determinate value for the limit. This method is particularly useful for limits that result in indeterminate forms. Remember to follow the steps outlined above to solve limits by factoring effectively.