Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

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Question: Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

Options:

Correct Answer: 7

Exam Year: 2021

Solution:

Factoring gives (x - 2)(x + 5)/(x - 2). For x ≠ 2, this simplifies to x + 5. Evaluating at x = 2 gives 7.

Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

Step 1: Identify the limit we need to find: lim (x -> 2) (x^2 + 3x - 10)/(x - 2).
Step 2: Factor the numerator (x^2 + 3x - 10).
Step 3: To factor, look for two numbers that multiply to -10 and add to 3. These numbers are 5 and -2.
Step 4: Rewrite the numerator as (x - 2)(x + 5).
Step 5: Now, substitute this back into the limit: lim (x -> 2) ((x - 2)(x + 5))/(x - 2).
Step 6: Notice that (x - 2) in the numerator and denominator can be canceled out, as long as x is not equal to 2.
Step 7: This simplifies the limit to lim (x -> 2) (x + 5).
Step 8: Now, substitute x = 2 into the simplified expression (x + 5).
Step 9: Calculate 2 + 5, which equals 7.
Step 10: Therefore, the limit is 7.

Limits – Understanding how to evaluate the limit of a function as it approaches a specific value.
Factoring – The ability to factor polynomials to simplify expressions before evaluating limits.
Indeterminate Forms – Recognizing and resolving indeterminate forms, such as 0/0, that can occur in limit problems.