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MathCalculus

Limit Laws: Evaluate lim(x→1/2) (2x²+3x−2)/(2x−1)

Identify the 0/0 form, factor the numerator, and evaluate the rational-function limit as x approaches 1/2.

Question

Use direct substitution to show that lim⁡x→1/22x2+3x−22x−1\lim_{x\to1/2}\frac{2x^2+3x-2}{2x-1} initially has the indeterminate form 0/00/0. Then evaluate the limit.

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Evidence boundary

The expression and approach point are from OpenStax Exercise 101. The cancellation is valid only for x≠1/2x\ne1/2; the value of the original quotient at x=1/2x=1/2 is undefined, even though the limit exists.