Question Video: Finding the One-Sided Limits of a Function | Nagwa Question Video: Finding the One-Sided Limits of a Function | Nagwa

# Question Video: Finding the One-Sided Limits of a Function Mathematics • Second Year of Secondary School

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Determine lim_(π₯ β β9β») π(π₯) and lim_(π₯ β β9βΊ) π(π₯) given π(π₯) = {π₯ + 9, if π₯ β€ β9 and 1/(π₯ + 9), if π₯ > β9.

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### Video Transcript

Determine the left-sided limit as π₯ approaches negative nine of π of π₯ and the right-sided limit as π₯ approaches negative nine of π of π₯ given that π of π₯ is equal to π₯ plus nine if π₯ is less than or equal to negative nine and one over π₯ plus nine if π₯ is greater than negative nine.

Here, weβve been given a function defined piecewise over two intervals. For the left-sided limit, weβre approaching π₯ equals negative nine from the negative direction. Hence, π₯ is less than negative nine. For the right-sided limit, weβre approaching π₯ equals negative nine from the positive direction. And hence, π₯ is greater than negative nine. Since π₯ equals negative nine is the point between the two intervals of our piecewise function, for our left-sided limit will be in the first interval and for our right-sided limit will be in the second interval. Letβs work on finding the left-sided limit.

In this case, our function π of π₯ is π₯ plus nine. We can find this limit by direct substitution of π₯ equals negative nine into our function. Doing so, we find that our answer is negative nine plus nine which is equal to zero. The left-sided limit as π₯ approaches negative nine of π of π₯ is therefore zero. Now for the right-sided limit, here our function π of π₯ is one over π₯ plus nine. Again, we tried direct substitution of π₯ equals negative nine into our function. This time, doing so gives us an answer of one over zero. And as we know, dividing one by zero cannot be evaluated to a numerical value. In cases such as this, we say that the limit does not exist. And so in a strict sense, this is the answer to our question.

To more fully understand our result, however, letβs look at a graphical plot of our function. Here, we have sketched our graph. And we know that, in the interval where π₯ is less than or equal to negative nine, we have a well behaved function. And we know that due to the solid dot here at negative nine π₯ is indeed defined at this point. For the other interval, we know that as π₯ approaches negative nine, we have a vertical asymptote. This means that the values of π of π₯ get arbitrarily large. And this is often represented as infinity. In this sense, it is common to write that the right-sided limit as π₯ approaches negative nine of π of π₯ is equal to positive infinity.

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