Evaluate each limit.
lim x→2 √4x+10 / 2x−2

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Identify the limit expression: \( \lim_{{x \to 2}} \frac{\sqrt{4x+10}}{2x-2} \).

Substitute \( x = 2 \) into the expression to check if it results in an indeterminate form.

Notice that substituting \( x = 2 \) gives \( \frac{\sqrt{4(2)+10}}{2(2)-2} = \frac{\sqrt{18}}{2} \), which is not an indeterminate form.

Simplify the expression \( \frac{\sqrt{18}}{2} \) by recognizing that \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \).

Conclude that the limit is \( \frac{3\sqrt{2}}{2} \).

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주요 개념

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as x approaches 2.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In the given limit, the expression involves a square root in the numerator and a linear polynomial in the denominator. Understanding how to simplify or manipulate these functions is essential for evaluating limits, especially when direct substitution leads to indeterminate forms.

Substitution and Simplification

Substitution and simplification are techniques used in calculus to evaluate limits. When direct substitution results in an indeterminate form, such as 0/0, it is often necessary to simplify the expression or use algebraic manipulation to find the limit. This may involve factoring, rationalizing, or applying L'Hôpital's Rule to resolve the limit effectively.

Evaluate each limit. lim x→2 √4x+10 / 2x−2