Find the value of a constant that makes a piecewise function continuous by evaluating a limit using the conjugate method.

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Making a Piecewise Function Continuous Using Rationalization

Continuity core

For what value of $k$ is the following a continuous function? \[ f(x) = \begin{cases} \frac{\sqrt{7x+2}-\sqrt{6x+4}}{x-2} & \text{if } x \ge -\frac{2}{7} \text{ and } x \ne 2 \\ k & \text{if } x = 2 \end{cases} \]

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What you need first

Evaluating limits by direct substitution
Recognizing the 0/0 indeterminate form
Multiplying by a conjugate to rationalize a numerator
Simplifying algebraic fractions

What this problem tests

Tests the ability to apply the definition of continuity at a point and to resolve a 0/0 indeterminate form by multiplying by a conjugate.

This is a standard Calculus I problem testing the definition of continuity and the algebraic evaluation of limits involving radicals.

Making a Piecewise Function Continuous Using Rationalization

What you need first

What this problem tests

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