Limits and L'Hospital's Rule

1. L'Hospital's Rule 1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x\to 0} {f(x) \over g(x)} = lim_{x\to 0} {f'(x) \over g'(x)} }
2. L'Hospital's Rule 2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x\to \infty } {f(x) \over g(x)} = lim_{x\to \infty} {f'(x) \over g'(x)} }

there are some of the problems that can't use the L'Hospital's Rule such as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {x + cos x \over x}}

If you are only using the L'Hospital's Rule this is what you will get:

Start Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {x + cos x \over x}}

1. DerivativesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {1 - sin x \over 1}}
2. DerivativesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {- cos x}}
3. DerivativesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {sin x}}

And then so on and forth so that means the limit does not exist.

So if you wan't to compute this first limit we can't use L'Hospital's Rule and this is how we do it.

Start Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x \to \infty } {x + cos x \over x}}

1. <math display=inline> \lim_{x \to \infty } {1 {cos \over x}}</math