Evaluate limit x->2 (from the left) : f(x)= (x-3)/(x^2-4)?

Question

I know how to evaluate limits at infinity but I don't know how to evaluate it at a number. Can someone give show me the answer and explain why it is. THANKS

Answer 1

You can do it a couple of ways:

1) simply draw up a table
x.....1.9.....1.99.......1.999
f(x) ..2.8.....25.3.......250.3

etc which shows you that as you approach 2 from below (from the left) f(x) approaches positive infinity - note the same goes to working from the right - and you do sometimes get different answers.

2) (x-3)/(x^2-4) = (x-3)/[(x-2)(x+2)]
= [(x-3)/(x+2) * (1/(x-2)]
Now (x-3)/(x+2) is not an issue and will tend to -1/4 as x tends to 2 but it is known that 1/(x-2) tends to minus infinity from the left and plus infinity to the right (in the same way the 1/x does from 0) - so f(x) must therefore tend to -1/4 *-infinity from the left
= infinity (positive) - as seen from the table

Answer 2

As x→2 from below, f(x) increases without bound. ∞ is one way to say it. More accurately, the limit does not exist. The function has vertical asymptotes at x = ±2, an x intercept at x=3, a y intercept at y = 3/4, and a horizontal asymptote at y = 0. On the left, f approaches 0 from below, on the right of x=3, it approaches 0 from above.

Answer 3

Set the function equivalent to a different variable, say "y": f(x) = y y = (x^2 - 4) / (2x^2) Now sparkling up this for x in terms of y. you will get a sparkling function in terms of y. This function stands out as the inverse of f(x). the least confusing thank you to do it is to in all possibility merely multiply the two facets by using 2x^2 and combine like words. you may ultimately get: (2y - a million)*x^2 = 4 x^2 = 4 / (2y-a million) x = sqrt(4 / (2y-a million)) and so forth. So the inverse function of f(x), enable's call it g(y), is: g(y) = (2 * sqrt(2y-a million)) / (2y-a million)

Answer 4

limit x->2 (from the left) : f(x)= (x-3)/(x^2-4) = ∞
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Ideas x-3 -> -1, and x^2-4 -> 0-