Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).

₹0.0

Question: Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).

Options:

Correct Answer: 2

Solution:

Factoring gives (x-2)(x+2)/(x-2). Canceling gives lim x->2 (x + 2) = 4.

Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).

Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).

Correct Answer: 4

Step 1: Identify the limit you need to evaluate: lim x->2 (x^2 - 4)/(x - 2).
Step 2: Notice that the expression (x^2 - 4) can be factored. It is a difference of squares.
Step 3: Factor (x^2 - 4) into (x - 2)(x + 2).
Step 4: Rewrite the limit with the factored expression: lim x->2 ((x - 2)(x + 2))/(x - 2).
Step 5: You can see that (x - 2) is in both the numerator and the denominator.
Step 6: Cancel (x - 2) from the numerator and the denominator, which simplifies the expression to lim x->2 (x + 2).
Step 7: Now, substitute x = 2 into the simplified expression (x + 2).
Step 8: Calculate (2 + 2) which equals 4.
Step 9: Therefore, the limit is 4.

Limit Evaluation – Understanding how to evaluate limits, particularly when direct substitution leads to an indeterminate form.
Factoring – Using algebraic manipulation, such as factoring, to simplify expressions before evaluating limits.
Cancellation of Terms – Recognizing when and how to cancel common factors in a limit expression to resolve indeterminate forms.