Chapter 14: Q. 10 (page 889)

True or False For any function $f$ , and any real number c, $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x)$ .

Short Answer

Expert verified

The statement is false.

Step by step solution

Step. 1 Given Information

It is given in the question that $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x)$ , at any c.

i.e., Left hand limit is equal to right hand limit for the function $f (x) a t x = c .$

Step. 2 Example

Let suppose this function,

$f (x) = \{\begin{cases} 2 x + 1, x < 0 \\ 0, x ⩾ 0 \end{cases}$

Now, $\lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{-}} (2 x + 1) = 2 (0) + 1 = 1, \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{+}} 0 = 0,$

We can see clearly from this example that LHL is not equal to RHL.

Step. 3 Conclusion

From above example it is clear that there exist atleast one function for which LHL is not equal to RHL. Therefore, the above statement is false.

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True or False For any function $f$ , and any real number c, $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x)$ .

Short Answer

Step by step solution

Step. 1 Given Information

Step. 2 Example

Step. 3 Conclusion

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True or False For any function f, and any real number c,limx→c-f(x)=limx→c+f(x).

Short Answer

Step by step solution

Step. 1 Given Information

Step. 2 Example

Step. 3 Conclusion

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True or False For any function $f$ , and any real number c, $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x)$ .