Find the value of a constant that makes a piecewise rational function continuous by evaluating its limit at the point of discontinuity.

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Finding a Constant to Make a Piecewise Function Continuous

Continuity warmup

If the function \[ f(x) = \begin{cases} \frac{x^2-16}{x-4} & \text{if } x \ne 4 \\ C & \text{if } x = 4 \end{cases} \] is continuous, what is the value of \( C \)?

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What you need first

Evaluating limits of rational functions
Factoring polynomials
Understanding piecewise functions

What this problem tests

Tests the understanding that continuity requires the limit to equal the function value, and the algebraic skill to evaluate limits by factoring.

This is a standard introductory calculus problem testing the definition of continuity and the evaluation of limits resulting in a 0/0 indeterminate form.

Finding a Constant to Make a Piecewise Function Continuous

What you need first

What this problem tests

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