The limits are (a) 1 and (b) 1 because the question is asking for the limits of the function h which is the vertical position of the curve (usually called the y coordinate). The curve is continuous, no breaks, so you can just plug in values which tell you where to look along the horizontal axis. You then go straight up from these positions till you meet the curve to find the vertical coordinates which are values of h corresponding to the horizontal positions. It's these vertical positions that are the limits required.
Just to make it clear, in this question the value of x is used to find what goes in the parentheses with h, that is, h(something). But you have to evaluate the expression in the parentheses first (plug x into the expression and evaluate the result) to get the horizontal coordinate. The question is testing whether you confuse the horizontal position (normally called the x coordinate) with the value of x as given (limit as x approaches a certain value). In this case x is used as a "dummy" variable to plug a value into an expression and it's the expression which is used to determine the horizontal position. So effectively the expression (not x itself) is the x (horizontal) coordinate you need to look at on the graph. If you're confused, that's just the question trying to trick you!