Good time of day. There was a problem while researching the function graph. Namely, when finding the inclined asymptote. Graph: y=x-ln(x+1)
When calculating the limit, the coefficient k = 1. Calculate the limit with the formula under b and it turns out that b = minus infinity. I've counted it many times, and I don't understand what's the matter. I decided to check the graph using graphical calculators, and it is clearly shifted to the bottom. Please help me!
Well, it can happen that the limit $\lim_{x\to \infty}{\frac{f(x)}{x}}$ for the slope is finite, but contrariwise the intercept's one $\lim_{x\to \infty}{(f(x)-kx)}$ is not finite or it doesn't even exist (think about $x+cos(x)$). To have a so called oblique asymptote, both limits must exist and be finite. In your case, the function's derivative/slope tend to a constant as x approaches infinity, but the intercept doesn't converge, thus no oblique asymptote. In fact, if you graph a line $y = x + q$, you will see that, however low the intercept $q$ is, it will always intersect the function.