What is the lim as x--->2 of g(f(x)) with reference to this diagram:
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My reasoning thus far is that there is no limit because as x approaches 2 in the first graph, f(x) approaches 0. But in the second graph, of g(x), the function is not continuous at x = 0.
Is it DNE?
Right, the problem does not specify whether x is approaching 2 from the left or the right side. On the graph of f(x) it approaches 0 from both sides, but if we approach zero on the graph of g(x), g(x) is different depending on whether we approach from the left or right side.
Sorry, I meant to say that on the graph of f(x) as x approaches 2 from both sides, f(x) approaches zero. However, on the graph of g(x) you cannot approach 0 from both sides with the limit being the same. Is the correct way of interpreting the question to check the value that f(x) approaches as x approaches 2 on the graph of f(x) and then take that value in order to examine what value(s) g(x) assumes?
Thanks a lot! I got the answer. I never looked at it from that perspective, but since f(x) > 0 in from both the left and right side, we must approach the value g(x) thinking that the x-axis is actually the f(x) axis, meaning that we approaching f(x) = 0 from its positive side.
Hey all,
This question is free game for anyone. In regards to the diagram referenced in the first post, to which I posted a link, is the value of c that makes [f(x) + c g(x)] exist equal to the same value of c that makes f(x) + c g(x) continuous at x = 1.
I approached the problem this way:
For to exist, both the left and right limits must exist.
(The question mark is there not because this limit is unknown, but I leave it to you to figure it out.) Also, the left and right limits must be equal to each other. After finding the second limit, set the two of them equal to each other, and solve for c.
Now refer to the definition of continuity to complete the question, and proceed in a manner similar to above.