Suppose p and q are polynomials. If lim x→0 p(x) / q(x)=10 and q(0)=2, find p(0).

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Identify the given limit: $ \lim_{x \to 0} \frac{p(x)}{q(x)} = 10 $.

Recognize that $ q(0) = 2 $ is given, which means $ q(x) $ is not zero at $ x = 0 $.

Apply the limit definition: $ \lim_{x \to 0} \frac{p(x)}{q(x)} = \frac{p(0)}{q(0)} $.

Substitute the known values into the limit equation: $ \frac{p(0)}{2} = 10 $.

Solve for $ p(0) $ by multiplying both sides by 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. The notation lim x→0 p(x) / q(x) indicates the limit of the ratio of two polynomials p(x) and q(x) as x approaches 0. Understanding limits is crucial for evaluating expressions that may be indeterminate at specific points.

Polynomial Functions

Polynomials are mathematical expressions consisting of variables raised to non-negative integer powers and their coefficients. They can be represented in the form p(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0. The behavior of polynomials near specific points, such as x=0, is essential for evaluating limits and understanding continuity.

Continuity and Evaluation at a Point

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this case, since lim x→0 p(x) / q(x) = 10 and q(0) = 2, we can use the limit to find p(0) by rearranging the limit expression, leading to p(0) = 10 * q(0).

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