# Given #(x−4)/ (x^2+6x−40)# how do you find the limit as x approaches 4?

But we might also have spotted that:

And

I was slightly surprised not to find a two-sided limit here, as they often come out of these kinds of question. But the fact that L'Hôpital's Rule led directly to the solution, as opposed to being one step off the two-sided solution, was the spur to factorise.

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To find the limit as x approaches 4 for the expression (x−4)/ (x^2+6x−40), we can substitute the value of 4 into the expression and simplify. By substituting 4 for x, we get (4−4)/ (4^2+6(4)−40), which simplifies to 0/0. This is an indeterminate form. To evaluate the limit, we can factor the denominator and cancel out the common factor of (x−4). Factoring the denominator gives us (x−4)(x+10), and canceling out (x−4) from the numerator and denominator leaves us with 1/(x+10). Now, we can substitute 4 into the simplified expression, which gives us 1/(4+10) = 1/14. Therefore, the limit as x approaches 4 for the given expression is 1/14.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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