Answer
a) 0
b) Does not exist
c) 0
d) 3
e) 0
f) Limit does not exist
g) Limit does not exist
h) 1
Work Step by Step
a) $\lim\limits_{x\to 2}f(x)$ + $\lim\limits_{x \to 2}g(x)$ = 0+0 = 0
The above solution means that we need to approach 2 in both graphs and when we approach 2 in both graphs it turns out to be 0 so when we add two zeros we get 0 in return.
b) $\lim\limits_{x\to 0}f(x)$ + $\lim\limits_{x \to 0}g(x)$ = does not exist + 2 = Does not exist
In above problem when we have to approach 0 in both the graphs. For f(x) we will use first graph and we can see that when we approach from left side it gives 0 however from right side it gives -2 so as both values are not same so limit does not exist now for g(x) we will use second graph and we can see that when we approach from left and right side it gives 2 but when we add both the answers as our first limit does not exist so our limit in our final answers does not exist.
c) $\lim\limits_{x\to 0+}f(x)$ + $\lim\limits_{x \to 0+}g(x)$ = -2 + 2 = 0
In the above problem we need to approach 0 from positive side by seeing at the graph of f(x) we can conclude that on approaching from positive side we get -2 and by looking at graph of g(x) we can conclude that on approaching from positive side we get +2 so when we add -2 and +2 we get 0.
d) $\lim\limits_{x\to 0-}f(x)$ + $\lim\limits_{x \to 0-}g(x)$ = 1+2 = 3
we need to approach 0 from negative side by seeing at the graph of f(x) we can conclude that on approaching from negative side we get 1 and by looking at graph of g(x) we can conclude that on approaching from negative side we get 2 so when we add 1 and 2 we get 3.
e) $\lim\limits_{x\to 2}f(x)/(1+g(x))$ = 0/(1+0) = 0
In the above problem we have numerator as well as denominator by checking limit in numerator and by looking at graph of f(x) we can see that by approaching 2 from both sides we get 0 and denominator has g(x) by looking at graph of g(x) we conclude that on approaching 2 from both sides we get 0 as well on solving further we get 0 as a result .
f) $\lim\limits_{x\to 2}(1+g(x))/f(x)$ = 1/0 = Does not exist
we have numerator and denominators in the above problem if we look at numerator we have 1+ g(x) by looking at the graph of g(x) and approaching 2 from both sides we get 0 and answer for numerator becomes 1 but if we look at denominator we have only f(x) and after looking at graph of f(x) and approaching it from both sides we get 0 and as denominator has 0 so solution does not exist.
g) $\lim\limits_{x \to 0+} \sqrt f(x)$ = $\sqrt -2$ = does not exist
in the above problem when we look at graph of f(x) and approach f(x) from positive 0 we get -2 as a result but we cannot take square root of a negative value so as a result the limit does not exist.
h) $\lim\limits_{x \to 0-} \sqrt f(x)$ = $\sqrt 1$ = 1
In the above problem when we look at the graph of f(x) and approach f(x) from negative 0 side we get 1 in result which is a positive value and when we take square root of it we get 1 so the answer is 1 .
