Question Video: Finding the Limit of Rational Functions at a Point Mathematics • Higher Education

Determine lim as 𝑥 → 0 of (5𝑥² + 7𝑥)/3𝑥.

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Video Transcript

Determine the limit as 𝑥 approaches zero of five 𝑥 squared plus seven 𝑥 all divided by three 𝑥.

In this question, we’re asked to evaluate the limit of a rational function. That’s the quotient of two polynomials. And we recall we can always attempt to evaluate the limit of a rational function by direct substitution. However, if we substitute 𝑥 is equal to zero into our rational function and evaluate, we end up with zero divided by zero. This is an indeterminate form. This means we can’t evaluate this limit by direct substitution alone. We’re going to need to use some other method. Instead, let’s try and simplify our limit. And the first thing we could notice is there’s a shared factor of 𝑥 in the numerator and denominator. We want to cancel this shared factor of 𝑥 to simplify our limit.

And in fact, we’re allowed to do this. This is because we’re taking the limit as 𝑥 approaches zero. Remember, this means we’re interested in what happens to the outputs of our functions as our values of 𝑥 approach zero. Therefore, the output of the function when 𝑥 is equal to zero does not affect its limit as 𝑥 approaches zero. Another way of thinking about this is if 𝑥 is not equal to zero, 𝑥 divided by 𝑥 is just equal to one. Therefore, after canceling the shared factor of 𝑥 in the numerator and denominator, we’ve rewritten our limit as the limit as 𝑥 approaches zero of five 𝑥 plus seven all divided by three.

And now this is just a linear function, so we can evaluate this limit by using direct substitution. We substitute 𝑥 is equal to zero into our function to get five times zero plus seven all divided by three, which, if we simplify, is equal to seven over three, which is our final answer. Therefore, we were able to show the limit as 𝑥 approaches zero of five 𝑥 squared plus seven 𝑥 all divided by three 𝑥 is seven over three.

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